Bootstrapping (statistics)

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class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Disadvantages</span> </div> </a> <ul id="toc-Disadvantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recommendations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recommendations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Recommendations</span> </div> </a> <ul id="toc-Recommendations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_bootstrap_scheme" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types_of_bootstrap_scheme"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of bootstrap scheme</span> </div> </a> <button aria-controls="toc-Types_of_bootstrap_scheme-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of bootstrap scheme subsection</span> </button> <ul id="toc-Types_of_bootstrap_scheme-sublist" class="vector-toc-list"> <li id="toc-Case_resampling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Case_resampling"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Case resampling</span> </div> </a> <ul id="toc-Case_resampling-sublist" class="vector-toc-list"> <li id="toc-Estimating_the_distribution_of_sample_mean" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Estimating_the_distribution_of_sample_mean"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Estimating the distribution of sample mean</span> </div> </a> <ul id="toc-Estimating_the_distribution_of_sample_mean-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regression" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Regression</span> </div> </a> <ul id="toc-Regression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bayesian_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bayesian_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Bayesian bootstrap</span> </div> </a> <ul id="toc-Bayesian_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Smooth_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Smooth_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Smooth bootstrap</span> </div> </a> <ul id="toc-Smooth_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametric_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parametric_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Parametric bootstrap</span> </div> </a> <ul id="toc-Parametric_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resampling_residuals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resampling_residuals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Resampling residuals</span> </div> </a> <ul id="toc-Resampling_residuals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussian_process_regression_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gaussian_process_regression_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Gaussian process regression bootstrap</span> </div> </a> <ul id="toc-Gaussian_process_regression_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wild_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wild_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Wild bootstrap</span> </div> </a> <ul id="toc-Wild_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Block_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Block_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Block bootstrap</span> </div> </a> <ul id="toc-Block_bootstrap-sublist" class="vector-toc-list"> <li id="toc-Time_series:_Simple_block_bootstrap" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time_series:_Simple_block_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.1</span> <span>Time series: Simple block bootstrap</span> </div> </a> <ul id="toc-Time_series:_Simple_block_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_series:_Moving_block_bootstrap" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time_series:_Moving_block_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.2</span> <span>Time series: Moving block bootstrap</span> </div> </a> <ul id="toc-Time_series:_Moving_block_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_series:_Maximum_entropy_bootstrap" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time_series:_Maximum_entropy_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.3</span> <span>Time series: Maximum entropy bootstrap</span> </div> </a> <ul id="toc-Time_series:_Maximum_entropy_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cluster_data:_block_bootstrap" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cluster_data:_block_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.4</span> <span>Cluster data: block bootstrap</span> </div> </a> <ul id="toc-Cluster_data:_block_bootstrap-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Methods_for_improving_computational_efficiency" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Methods_for_improving_computational_efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Methods for improving computational efficiency</span> </div> </a> <button aria-controls="toc-Methods_for_improving_computational_efficiency-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Methods for improving computational efficiency subsection</span> </button> <ul id="toc-Methods_for_improving_computational_efficiency-sublist" class="vector-toc-list"> <li id="toc-Parallel_processing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parallel_processing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Parallel processing</span> </div> </a> <ul id="toc-Parallel_processing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Poisson bootstrap</span> </div> </a> <ul id="toc-Poisson_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bag_of_Little_Bootstraps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bag_of_Little_Bootstraps"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Bag of Little Bootstraps</span> </div> </a> <ul id="toc-Bag_of_Little_Bootstraps-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Choice_of_statistic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Choice_of_statistic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Choice of statistic</span> </div> </a> <ul id="toc-Choice_of_statistic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deriving_confidence_intervals_from_the_bootstrap_distribution" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Deriving_confidence_intervals_from_the_bootstrap_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Deriving confidence intervals from the bootstrap distribution</span> </div> </a> <button aria-controls="toc-Deriving_confidence_intervals_from_the_bootstrap_distribution-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Deriving confidence intervals from the bootstrap distribution subsection</span> </button> <ul id="toc-Deriving_confidence_intervals_from_the_bootstrap_distribution-sublist" class="vector-toc-list"> <li id="toc-Desirable_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Desirable_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Desirable properties</span> </div> </a> <ul id="toc-Desirable_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bias,_asymmetry,_and_confidence_intervals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bias,_asymmetry,_and_confidence_intervals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Bias, asymmetry, and confidence intervals</span> </div> </a> <ul id="toc-Bias,_asymmetry,_and_confidence_intervals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods_for_bootstrap_confidence_intervals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methods_for_bootstrap_confidence_intervals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Methods for bootstrap confidence intervals</span> </div> </a> <ul id="toc-Methods_for_bootstrap_confidence_intervals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bootstrap_hypothesis_testing" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bootstrap_hypothesis_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bootstrap hypothesis testing</span> </div> </a> <ul id="toc-Bootstrap_hypothesis_testing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Example_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Example applications</span> </div> </a> <button aria-controls="toc-Example_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Example applications subsection</span> </button> <ul id="toc-Example_applications-sublist" class="vector-toc-list"> <li id="toc-Smoothed_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Smoothed_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Smoothed bootstrap</span> </div> </a> <ul id="toc-Smoothed_bootstrap-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_other_approaches_to_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_other_approaches_to_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Relation to other approaches to inference</span> </div> </a> <button aria-controls="toc-Relation_to_other_approaches_to_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation to other approaches to inference subsection</span> </button> <ul id="toc-Relation_to_other_approaches_to_inference-sublist" class="vector-toc-list"> <li id="toc-Relationship_to_other_resampling_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_other_resampling_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Relationship to other resampling methods</span> </div> </a> <ul id="toc-Relationship_to_other_resampling_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-U-statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#U-statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>U-statistics</span> </div> </a> <ul id="toc-U-statistics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Asymptotic_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Asymptotic_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Asymptotic theory</span> </div> </a> <button aria-controls="toc-Asymptotic_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Asymptotic theory subsection</span> </button> <ul id="toc-Asymptotic_theory-sublist" class="vector-toc-list"> <li id="toc-Stochastic_convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Stochastic convergence</span> </div> </a> <ul id="toc-Stochastic_convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Consistency</span> </div> </a> <ul id="toc-Consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strong_consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strong_consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Strong consistency</span> </div> </a> <ul id="toc-Strong_consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Showing_consistency_using_the_central_limit_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Showing_consistency_using_the_central_limit_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Showing consistency using the central limit theorem</span> </div> </a> <ul id="toc-Showing_consistency_using_the_central_limit_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Bootstrapping (statistics)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 23 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-23" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">23 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Bootstrapping_(estad%C3%ADstica)" title="Bootstrapping (estadística) – Catalan" lang="ca" hreflang="ca" data-title="Bootstrapping (estadística)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bootstrapping_(statistika)" title="Bootstrapping (statistika) – Czech" lang="cs" hreflang="cs" data-title="Bootstrapping (statistika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bootstrapping-Verfahren" title="Bootstrapping-Verfahren – German" lang="de" hreflang="de" data-title="Bootstrapping-Verfahren" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bootstrap_(statistika)" title="Bootstrap (statistika) – Estonian" lang="et" hreflang="et" data-title="Bootstrap (statistika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Bootstrapping_(estad%C3%ADstica)" title="Bootstrapping (estadística) – Spanish" lang="es" hreflang="es" data-title="Bootstrapping (estadística)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D9%88%D8%AA%E2%80%8C%D8%A7%D8%B3%D8%AA%D8%B1%D9%BE%DB%8C%D9%86%DA%AF_(%D8%A2%D9%85%D8%A7%D8%B1)" title="بوت‌استرپینگ (آمار) – Persian" lang="fa" hreflang="fa" data-title="بوت‌استرپینگ (آمار)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bootstrap_(statistiques)" title="Bootstrap (statistiques) – French" lang="fr" hreflang="fr" data-title="Bootstrap (statistiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%ED%8A%B8%EC%8A%A4%ED%8A%B8%EB%9E%A9_(%ED%86%B5%EA%B3%84%ED%95%99)" title="부트스트랩 (통계학) – Korean" lang="ko" hreflang="ko" data-title="부트스트랩 (통계학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pendayasahajaan_(statistik)" title="Pendayasahajaan (statistik) – Indonesian" lang="id" hreflang="id" data-title="Pendayasahajaan (statistik)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Bootstrap_(statistica)" title="Bootstrap (statistica) – Italian" lang="it" hreflang="it" data-title="Bootstrap (statistica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/Bootstrap_(%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94)" title="Bootstrap (סטטיסטיקה) – Hebrew" lang="he" hreflang="he" data-title="Bootstrap (סטטיסטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembutstrapan_(statistik)" title="Pembutstrapan (statistik) – Malay" lang="ms" hreflang="ms" data-title="Pembutstrapan (statistik)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%BC%E3%83%88%E3%82%B9%E3%83%88%E3%83%A9%E3%83%83%E3%83%97%E6%B3%95" title="ブートストラップ法 – Japanese" lang="ja" hreflang="ja" data-title="ブートストラップ法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bootstrapping" title="Bootstrapping – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bootstrapping" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Bootstrap_(statystyka)" title="Bootstrap (statystyka) – Polish" lang="pl" hreflang="pl" data-title="Bootstrap (statystyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Bootstrapping_(estat%C3%ADstica)" title="Bootstrapping (estatística) – Portuguese" lang="pt" hreflang="pt" data-title="Bootstrapping (estatística)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D1%83%D1%82%D1%81%D1%82%D1%80%D1%8D%D0%BF_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Бутстрэп (статистика) – Russian" lang="ru" hreflang="ru" data-title="Бутстрэп (статистика)" data-language-autonym="Русский" data-language-local-name="Russian" 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Statistical method</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Bootstrapping_(disambiguation)" class="mw-disambig" title="Bootstrapping (disambiguation)">Bootstrapping (disambiguation)</a>.</div> <p><b>Bootstrapping</b> is a procedure for estimating the distribution of an estimator by <a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">resampling</a> (often <a href="/wiki/Sampling_(statistics)#Replacement_of_selected_units" title="Sampling (statistics)">with replacement</a>) one's data or a model estimated from the data.<sup id="cite_ref-Horowitz2019_1-0" class="reference"><a href="#cite_note-Horowitz2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Bootstrapping assigns measures of accuracy (<a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a>, variance, <a href="/wiki/Confidence_interval" title="Confidence interval">confidence intervals</a>, prediction error, etc.) to sample estimates.<sup id="cite_ref-et1993_2-0" class="reference"><a href="#cite_note-et1993-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.<sup id="cite_ref-Horowitz2019_1-1" class="reference"><a href="#cite_note-Horowitz2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Bootstrapping estimates the properties of an <a href="/wiki/Estimand" title="Estimand">estimand</a> (such as its <a href="/wiki/Variance" title="Variance">variance</a>) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> of the observed data. In the case where a set of observations can be assumed to be from an <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> population, this can be implemented by constructing a number of <a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">resamples</a> with replacement, of the observed data set (and of equal size to the observed data set). A key result in Efron's seminal paper that introduced the bootstrap<sup id="cite_ref-Efron1979_4-0" class="reference"><a href="#cite_note-Efron1979-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> is the favorable performance of bootstrap methods using <a href="/wiki/Sampling_(statistics)#Replacement_of_selected_units" title="Sampling (statistics)">sampling with replacement</a> compared to prior methods like the <a href="/wiki/Jackknife_resampling" title="Jackknife resampling">jackknife</a> that sample without replacement. However, since its introduction, numerous variants on the bootstrap have been proposed, including methods that sample without replacement or that create bootstrap samples larger or smaller than the original data. </p><p>The bootstrap may also be used for constructing <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">hypothesis tests</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It is often used as an alternative to <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a> based on the assumption of a parametric model when that assumption is in doubt, or where parametric inference is impossible or requires complicated formulas for the calculation of <a href="/wiki/Standard_error" title="Standard error">standard errors</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> was first described by <a href="/wiki/Bradley_Efron" title="Bradley Efron">Bradley Efron</a> in "Bootstrap methods: another look at the jackknife" (1979),<sup id="cite_ref-Efron1979_4-2" class="reference"><a href="#cite_note-Efron1979-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> inspired by earlier work on the <a href="/wiki/Jackknife_resampling" title="Jackknife resampling">jackknife</a>.<sup id="cite_ref-Quenouille1949_7-0" class="reference"><a href="#cite_note-Quenouille1949-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tukey1958_8-0" class="reference"><a href="#cite_note-Tukey1958-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jaeckel1972_9-0" class="reference"><a href="#cite_note-Jaeckel1972-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Improved estimates of the variance were developed later.<sup id="cite_ref-Bickel1982_10-0" class="reference"><a href="#cite_note-Bickel1982-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Singh1981_11-0" class="reference"><a href="#cite_note-Singh1981-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A Bayesian extension was developed in 1981.<sup id="cite_ref-Rubin1981_12-0" class="reference"><a href="#cite_note-Rubin1981-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The bias-corrected and accelerated (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4202b26ffbdf0f6ac4b409a06ab6c615d46afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.528ex; height:2.509ex;" alt="{\displaystyle BC_{a}}"></span>) bootstrap was developed by Efron in 1987,<sup id="cite_ref-BCa_13-0" class="reference"><a href="#cite_note-BCa-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and the approximate bootstrap confidence interval (ABC, or approximate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4202b26ffbdf0f6ac4b409a06ab6c615d46afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.528ex; height:2.509ex;" alt="{\displaystyle BC_{a}}"></span>) procedure in 1992.<sup id="cite_ref-Diciccio1992_14-0" class="reference"><a href="#cite_note-Diciccio1992-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Approach">Approach</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=2" title="Edit section: Approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Illustration_bootstrap.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Illustration_bootstrap.svg/400px-Illustration_bootstrap.svg.png" decoding="async" width="400" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Illustration_bootstrap.svg/600px-Illustration_bootstrap.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Illustration_bootstrap.svg/800px-Illustration_bootstrap.svg.png 2x" data-file-width="1000" data-file-height="304" /></a><figcaption>A sample is drawn from a population. From this sample, resamples are generated by drawing with replacement (orange). Data points that were drawn more than once (which happens for approx. 26.4% of data points) are shown in red and slightly offsetted. From the resamples, the statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is calculated and, therefore, a histogram can be calculated to estimate the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</figcaption></figure> <p>The basic idea of bootstrapping is that inference about a population from sample data (sample → population) can be modeled by <i>resampling</i> the sample data and performing inference about a sample from resampled data (resampled → sample).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> As the population is unknown, the true error in a sample statistic against its population value is unknown. In bootstrap-resamples, the 'population' is in fact the sample, and this is known; hence the quality of inference of the 'true' sample from resampled data (resampled → sample) is measurable. </p><p>More formally, the bootstrap works by treating inference of the true <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> <i>J</i>, given the original data, as being analogous to an inference of the empirical distribution <i>Ĵ</i>, given the resampled data. The accuracy of inferences regarding <i>Ĵ</i> using the resampled data can be assessed because we know <i>Ĵ</i>. If <i>Ĵ</i> is a reasonable approximation to <i>J</i>, then the quality of inference on <i>J</i> can in turn be inferred. </p><p>As an example, assume we are interested in the average (or <a href="/wiki/Mean" title="Mean">mean</a>) height of people worldwide. We cannot measure all the people in the global population, so instead, we sample only a tiny part of it, and measure that. Assume the sample is of size <i>N</i>; that is, we measure the heights of <i>N</i> individuals. From that single sample, only one estimate of the mean can be obtained. In order to reason about the population, we need some sense of the <a href="/wiki/Statistical_dispersion" title="Statistical dispersion">variability</a> of the mean that we have computed. The simplest bootstrap method involves taking the original data set of heights, and, using a computer, sampling from it to form a new sample (called a 'resample' or bootstrap sample) that is also of size <i>N</i>. The bootstrap sample is taken from the original by using <a href="/wiki/Simple_random_sample" title="Simple random sample">sampling with replacement</a> (e.g. we might 'resample' 5 times from [1,2,3,4,5] and get [2,5,4,4,1]), so, assuming <i>N</i> is sufficiently large, for all practical purposes there is virtually zero probability that it will be identical to the original "real" sample. This process is repeated a large number of times (typically 1,000 or 10,000 times), and for each of these bootstrap samples, we compute its mean (each of these is called a "bootstrap estimate"). We now can create a histogram of bootstrap means. This histogram provides an estimate of the shape of the distribution of the sample mean from which we can answer questions about how much the mean varies across samples. (The method here, described for the mean, can be applied to almost any other <a href="/wiki/Statistic" title="Statistic">statistic</a> or <a href="/wiki/Estimator" title="Estimator">estimator</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Discussion">Discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=3" title="Edit section: Discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Advantages">Advantages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=4" title="Edit section: Advantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A great advantage of bootstrap is its simplicity. It is a straightforward way to derive estimates of <a href="/wiki/Standard_error_(statistics)" class="mw-redirect" title="Standard error (statistics)">standard errors</a> and <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> for complex estimators of the distribution, such as percentile points, proportions, <a href="/wiki/Odds_ratio" title="Odds ratio">Odds ratio</a>, and correlation coefficients. However, despite its simplicity, bootstrapping can be applied to complex sampling designs (e.g. for population divided into s strata with n<sub>s</sub> observations per strata, bootstrapping can be applied for each stratum).<sup id="cite_ref-ch21_brm_16-0" class="reference"><a href="#cite_note-ch21_brm-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Bootstrap is also an appropriate way to control and check the stability of the results. Although for most problems it is impossible to know the true confidence interval, bootstrap is asymptotically more accurate than the standard intervals obtained using sample variance and assumptions of normality.<sup id="cite_ref-DiCiccio1996_17-0" class="reference"><a href="#cite_note-DiCiccio1996-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Bootstrapping is also a convenient method that avoids the cost of repeating the experiment to get other groups of sample data. </p> <div class="mw-heading mw-heading3"><h3 id="Disadvantages">Disadvantages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=5" title="Edit section: Disadvantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bootstrapping depends heavily on the estimator used and, though simple, naive use of bootstrapping will not always yield asymptotically valid results and can lead to inconsistency.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Although bootstrapping is (under some conditions) asymptotically <a href="/wiki/Consistency_(statistics)" title="Consistency (statistics)">consistent</a>, it does not provide general finite-sample guarantees. The result may depend on the representative sample. The apparent simplicity may conceal the fact that important assumptions are being made when undertaking the bootstrap analysis (e.g. independence of samples or large enough of a sample size) where these would be more formally stated in other approaches. Also, bootstrapping can be time-consuming and there are not many available software for bootstrapping as it is difficult to automate using traditional statistical computer packages.<sup id="cite_ref-ch21_brm_16-1" class="reference"><a href="#cite_note-ch21_brm-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Recommendations">Recommendations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=6" title="Edit section: Recommendations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Scholars have recommended more bootstrap samples as available computing power has increased. If the results may have substantial real-world consequences, then one should use as many samples as is reasonable, given available computing power and time. Increasing the number of samples cannot increase the amount of information in the original data; it can only reduce the effects of random sampling errors which can arise from a bootstrap procedure itself. Moreover, there is evidence that numbers of samples greater than 100 lead to negligible improvements in the estimation of standard errors.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In fact, according to the original developer of the bootstrapping method, even setting the number of samples at 50 is likely to lead to fairly good standard error estimates.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Adèr et al. recommend the bootstrap procedure for the following situations:<sup id="cite_ref-Ader_21-0" class="reference"><a href="#cite_note-Ader-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><ul><li>When the theoretical distribution of a statistic of interest is complicated or unknown. Since the bootstrapping procedure is distribution-independent it provides an indirect method to assess the properties of the distribution underlying the sample and the parameters of interest that are derived from this distribution.</li></ul></dd></dl> <dl><dd><ul><li>When the <a href="/wiki/Sample_size" class="mw-redirect" title="Sample size">sample size</a> is insufficient for straightforward statistical inference. If the underlying distribution is well-known, bootstrapping provides a way to account for the distortions caused by the specific sample that may not be fully representative of the population.</li></ul></dd></dl> <dl><dd><ul><li>When <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">power calculations</a> have to be performed, and a small pilot sample is available. Most power and sample size calculations are heavily dependent on the standard deviation of the statistic of interest. If the estimate used is incorrect, the required sample size will also be wrong. One method to get an impression of the variation of the statistic is to use a small pilot sample and perform bootstrapping on it to get impression of the variance.</li></ul></dd></dl> <p>However, Athreya has shown<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> that if one performs a naive bootstrap on the sample mean when the underlying population lacks a finite variance (for example, a <a href="/wiki/Power_law_distribution" class="mw-redirect" title="Power law distribution">power law distribution</a>), then the bootstrap distribution will not converge to the same limit as the sample mean. As a result, confidence intervals on the basis of a <a href="/wiki/Monte_Carlo_simulation" class="mw-redirect" title="Monte Carlo simulation">Monte Carlo simulation</a> of the bootstrap could be misleading. Athreya states that "Unless one is reasonably sure that the underlying distribution is not <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy tailed</a>, one should hesitate to use the naive bootstrap". </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_bootstrap_scheme">Types of bootstrap scheme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=7" title="Edit section: Types of bootstrap scheme"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In univariate problems, it is usually acceptable to resample the individual observations with replacement ("case resampling" below) unlike <a href="/wiki/Resampling_(statistics)#Subsampling" title="Resampling (statistics)">subsampling</a>, in which resampling is without replacement and is valid under much weaker conditions compared to the bootstrap. In small samples, a parametric bootstrap approach might be preferred. For other problems, a <i>smooth bootstrap</i> will likely be preferred. </p><p>For regression problems, various other alternatives are available.<sup id="cite_ref-et1993_2-1" class="reference"><a href="#cite_note-et1993-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Case_resampling">Case resampling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=8" title="Edit section: Case resampling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap is generally useful for estimating the distribution of a statistic (e.g. mean, variance) without using normality assumptions (as required, e.g., for a z-statistic or a t-statistic). In particular, the bootstrap is useful when there is no analytical form or an asymptotic theory (e.g., an applicable <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>) to help estimate the distribution of the statistics of interest. This is because bootstrap methods can apply to most random quantities, e.g., the ratio of variance and mean. There are at least two ways of performing case resampling. </p> <ol><li>The Monte Carlo algorithm for case resampling is quite simple. First, we resample the data with replacement, and the size of the resample must be equal to the size of the original data set. Then the statistic of interest is computed from the resample from the first step. We repeat this routine many times to get a more precise estimate of the Bootstrap distribution of the statistic.<sup id="cite_ref-et1993_2-2" class="reference"><a href="#cite_note-et1993-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>The 'exact' version for case resampling is similar, but we exhaustively enumerate every possible resample of the data set. This can be computationally expensive as there are a total of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {2n-1}{n}}={\frac {(2n-1)!}{n!(n-1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {2n-1}{n}}={\frac {(2n-1)!}{n!(n-1)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864f94e4ee3ef9ab67ee4c4140ecd21d4d14f33b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.811ex; height:6.509ex;" alt="{\displaystyle {\binom {2n-1}{n}}={\frac {(2n-1)!}{n!(n-1)!}}}"></span> different resamples, where <i>n</i> is the size of the data set. Thus for <i>n</i> = 5, 10, 20, 30 there are 126, 92378, 6.89 × 10<sup>10</sup> and 5.91 × 10<sup>16</sup> different resamples respectively.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading4"><h4 id="Estimating_the_distribution_of_sample_mean">Estimating the distribution of sample mean</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=9" title="Edit section: Estimating the distribution of sample mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a coin-flipping experiment. We flip the coin and record whether it lands heads or tails. Let <span class="nowrap"><i>X = x</i><sub>1</sub>, <i>x</i><sub>2</sub>, …, <i>x</i><sub>10</sub></span> be 10 observations from the experiment. <span class="nowrap"><i>x<sub>i</sub></i> = 1</span> if the i th flip lands heads, and 0 otherwise. By invoking the assumption that the average of the coin flips is normally distributed, we can use the <a href="/wiki/Student%27s_t-statistic" class="mw-redirect" title="Student's t-statistic">t-statistic</a> to estimate the distribution of the sample mean, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}={\frac {1}{10}}(x_{1}+x_{2}+\cdots +x_{10}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}={\frac {1}{10}}(x_{1}+x_{2}+\cdots +x_{10}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d4303a595dc15e087045fc3ee65b96d59edc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.263ex; height:5.176ex;" alt="{\displaystyle {\bar {x}}={\frac {1}{10}}(x_{1}+x_{2}+\cdots +x_{10}).}"></span></dd></dl> <p>Such a normality assumption can be justified either as an approximation of the distribution of each <i>individual</i> coin flip or as an approximation of the distribution of the <i>average</i> of a large number of coin flips. The former is a poor approximation because the true distribution of the coin flips is <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a> instead of normal. The latter is a valid approximation in <i>infinitely large</i> samples due to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. </p><p>However, if we are not ready to make such a justification, then we can use the bootstrap instead. Using case resampling, we can derive the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></span>. We first resample the data to obtain a <i>bootstrap resample</i>. An example of the first resample might look like this <span class="nowrap"><i>X</i><sub>1</sub>* = <i>x</i><sub>2</sub>, <i>x</i><sub>1</sub>, <i>x</i><sub>10</sub>, <i>x</i><sub>10</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, <i>x</i><sub>6</sub>, <i>x</i><sub>7</sub>, <i>x</i><sub>1</sub>, <i>x</i><sub>9</sub></span>. There are some duplicates since a bootstrap resample comes from sampling with replacement from the data. Also the number of data points in a bootstrap resample is equal to the number of data points in our original observations. Then we compute the mean of this resample and obtain the first <i>bootstrap mean</i>: <i>μ</i><sub>1</sub>*. We repeat this process to obtain the second resample <i>X</i><sub>2</sub>* and compute the second bootstrap mean <i>μ</i><sub>2</sub>*. If we repeat this 100 times, then we have <i>μ</i><sub>1</sub>*, <i>μ</i><sub>2</sub>*, ..., <i>μ</i><sub>100</sub>*. This represents an <i>empirical bootstrap distribution</i> of sample mean. From this empirical distribution, one can derive a <i>bootstrap confidence interval</i> for the purpose of hypothesis testing. </p> <div class="mw-heading mw-heading4"><h4 id="Regression">Regression</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=10" title="Edit section: Regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In regression problems, <i>case resampling</i> refers to the simple scheme of resampling individual cases – often rows of a <a href="/wiki/Data_set" title="Data set">data set</a>. For regression problems, as long as the data set is fairly large, this simple scheme is often acceptable.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</sup> However, the method is open to criticism<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2009)">citation needed</span></a></i>]</sup>.<sup id="cite_ref-ch21_brm_16-2" class="reference"><a href="#cite_note-ch21_brm-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In regression problems, the <a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a> are often fixed, or at least observed with more control than the response variable. Also, the range of the explanatory variables defines the information available from them. Therefore, to resample cases means that each bootstrap sample will lose some information. As such, alternative bootstrap procedures should be considered. </p> <div class="mw-heading mw-heading3"><h3 id="Bayesian_bootstrap">Bayesian bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=11" title="Edit section: Bayesian bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bootstrapping can be interpreted in a <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian</a> framework using a scheme that creates new data sets through reweighting the initial data. Given a set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> data points, the weighting assigned to data point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in a new data set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}^{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}^{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545a8dd06a6d9d7252be893148969423e683e2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.065ex; height:2.676ex;" alt="{\displaystyle {\mathcal {D}}^{J}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}^{J}=x_{i}^{J}-x_{i-1}^{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}^{J}=x_{i}^{J}-x_{i-1}^{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b2a1e217dc002fe7ead5bba6be606ba3abd131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.708ex; height:3.343ex;" alt="{\displaystyle w_{i}^{J}=x_{i}^{J}-x_{i-1}^{J}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a955ad87471c98c487ba0a503029df754f60f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.684ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} ^{J}}"></span> is a low-to-high ordered list of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86aeb216b214f70df1341f34ce273cd3582ce2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-1}"></span> uniformly distributed random numbers on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span>, preceded by 0 and succeeded by 1. The distributions of a parameter inferred from considering many such data sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}^{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}^{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545a8dd06a6d9d7252be893148969423e683e2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.065ex; height:2.676ex;" alt="{\displaystyle {\mathcal {D}}^{J}}"></span> are then interpretable as <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distributions</a> on that parameter.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Smooth_bootstrap">Smooth bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=12" title="Edit section: Smooth bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under this scheme, a small amount of (usually normally distributed) zero-centered random noise is added onto each resampled observation. This is equivalent to sampling from a <a href="/wiki/Kernel_density" class="mw-redirect" title="Kernel density">kernel density</a> estimate of the data. Assume <i>K</i> to be a symmetric kernel density function with unit variance. The standard kernel estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f\,}}_{h}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f\,}}_{h}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a9b0f86f0f643bc2958e5d558307f3d73b0516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.984ex; height:3.343ex;" alt="{\displaystyle {\hat {f\,}}_{h}(x)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f\,}}_{h}(x)={1 \over nh}\sum _{i=1}^{n}K\left({x-X_{i} \over h}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f\,}}_{h}(x)={1 \over nh}\sum _{i=1}^{n}K\left({x-X_{i} \over h}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969dd5885616a07f74811cf15ecd75443c8c20fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.419ex; height:6.843ex;" alt="{\displaystyle {\hat {f\,}}_{h}(x)={1 \over nh}\sum _{i=1}^{n}K\left({x-X_{i} \over h}\right),}"></span> <sup id="cite_ref-wang1995_25-0" class="reference"><a href="#cite_note-wang1995-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is the smoothing parameter. And the corresponding distribution function estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {F\,}}_{h}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mspace width="thinmathspace" /> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {F\,}}_{h}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee1c0f6d4b01380ca94f2a457e004e043a5a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.446ex; height:3.343ex;" alt="{\displaystyle {\hat {F\,}}_{h}(x)}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {F\,}}_{h}(x)=\int _{-\infty }^{x}{\hat {f}}_{h}(t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mspace width="thinmathspace" /> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {F\,}}_{h}(x)=\int _{-\infty }^{x}{\hat {f}}_{h}(t)\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688597c3b198ee754f24f0fe70f465e3f83d569e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.994ex; height:6.009ex;" alt="{\displaystyle {\hat {F\,}}_{h}(x)=\int _{-\infty }^{x}{\hat {f}}_{h}(t)\,dt.}"></span> <sup id="cite_ref-wang1995_25-1" class="reference"><a href="#cite_note-wang1995-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Parametric_bootstrap">Parametric bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=13" title="Edit section: Parametric bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Based on the assumption that the original data set is a realization of a random sample from a distribution of a specific parametric type, in this case a parametric model is fitted by parameter θ, often by <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a>, and samples of <a href="/wiki/Random_number_generation" title="Random number generation">random numbers</a> are drawn from this fitted model. Usually the sample drawn has the same sample size as the original data. Then the estimate of original function F can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {F}}=F_{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {F}}=F_{\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9531f5db566d2e84111d327c4512e804fc0eadd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.646ex; height:3.676ex;" alt="{\displaystyle {\hat {F}}=F_{\hat {\theta }}}"></span>. This sampling process is repeated many times as for other bootstrap methods. Considering the centered <a href="/wiki/Sample_means" class="mw-redirect" title="Sample means">sample mean</a> in this case, the random sample original distribution function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8272c172045a658f8650af10412e76c9cfb0cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.498ex; height:2.509ex;" alt="{\displaystyle F_{\theta }}"></span> is replaced by a bootstrap random sample with function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7e81ed047ca9a0ca83d087c513826d5c4b9d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.742ex; height:3.009ex;" alt="{\displaystyle F_{\hat {\theta }}}"></span>, and the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X_{n}}}-\mu _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X_{n}}}-\mu _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6648afa6c5211e3faae8e578c6e6d58703144e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.388ex; height:3.009ex;" alt="{\displaystyle {\bar {X_{n}}}-\mu _{\theta }}"></span> is approximated by that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}^{*}-\mu ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}^{*}-\mu ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d800341f54241b0f14f38c0da0dd777b19a046ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.495ex; height:3.176ex;" alt="{\displaystyle {\bar {X}}_{n}^{*}-\mu ^{*}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{*}=\mu _{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{*}=\mu _{\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3b34e90697404871a1eac71fca9d95c6762e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.204ex; height:3.176ex;" alt="{\displaystyle \mu ^{*}=\mu _{\hat {\theta }}}"></span>, which is the expectation corresponding to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7e81ed047ca9a0ca83d087c513826d5c4b9d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.742ex; height:3.009ex;" alt="{\displaystyle F_{\hat {\theta }}}"></span>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The use of a parametric model at the sampling stage of the bootstrap methodology leads to procedures which are different from those obtained by applying basic statistical theory to inference for the same model. </p> <div class="mw-heading mw-heading3"><h3 id="Resampling_residuals">Resampling residuals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=14" title="Edit section: Resampling residuals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another approach to bootstrapping in regression problems is to resample <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">residuals</a>. The method proceeds as follows. </p> <ol><li>Fit the model and retain the fitted values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y\,}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y\,}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34970b4d78ab39658093762c96ba487c29f5cd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.342ex; height:2.843ex;" alt="{\displaystyle {\widehat {y\,}}_{i}}"></span> and the residuals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\varepsilon \,}}_{i}=y_{i}-{\widehat {y\,}}_{i},(i=1,\dots ,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ε</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\varepsilon \,}}_{i}=y_{i}-{\widehat {y\,}}_{i},(i=1,\dots ,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d339fdf06958aa88b5918ceedf9992cefb48d99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.97ex; height:2.843ex;" alt="{\displaystyle {\widehat {\varepsilon \,}}_{i}=y_{i}-{\widehat {y\,}}_{i},(i=1,\dots ,n)}"></span>.</li> <li>For each pair, (<i>x<sub>i</sub></i>, <i>y<sub>i</sub></i>), in which <i>x<sub>i</sub></i> is the (possibly multivariate) explanatory variable, add a randomly resampled residual, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\varepsilon \,}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ε</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\varepsilon \,}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7f275897d4ad1a194c18c86d49705c56c5994e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.38ex; height:3.009ex;" alt="{\displaystyle {\widehat {\varepsilon \,}}_{j}}"></span>, to the fitted value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y\,}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y\,}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34970b4d78ab39658093762c96ba487c29f5cd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.342ex; height:2.843ex;" alt="{\displaystyle {\widehat {y\,}}_{i}}"></span>. In other words, create synthetic response variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ε</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e354e6bec213e42443c1d048f6904f91448676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.876ex; height:3.009ex;" alt="{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{j}}"></span> where <i>j</i> is selected randomly from the list (1, ..., <i>n</i>) for every <i>i</i>.</li> <li>Refit the model using the fictitious response variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b358e19039c3610c53a3e38a67e3a4509561b3dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.215ex; height:2.843ex;" alt="{\displaystyle y_{i}^{*}}"></span>, and retain the quantities of interest (often the parameters, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981df89acc9152887d0477571cfd038cfa46ad7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.466ex; height:3.009ex;" alt="{\displaystyle {\widehat {\mu }}_{i}^{*}}"></span>, estimated from the synthetic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b358e19039c3610c53a3e38a67e3a4509561b3dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.215ex; height:2.843ex;" alt="{\displaystyle y_{i}^{*}}"></span>).</li> <li>Repeat steps 2 and 3 a large number of times.</li></ol> <p>This scheme has the advantage that it retains the information in the explanatory variables. However, a question arises as to which residuals to resample. Raw residuals are one option; another is <a href="/wiki/Studentized_residuals" class="mw-redirect" title="Studentized residuals">studentized residuals</a> (in linear regression). Although there are arguments in favor of using studentized residuals; in practice, it often makes little difference, and it is easy to compare the results of both schemes. </p> <div class="mw-heading mw-heading3"><h3 id="Gaussian_process_regression_bootstrap">Gaussian process regression bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=15" title="Edit section: Gaussian process regression bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When data are temporally correlated, straightforward bootstrapping destroys the inherent correlations. This method uses Gaussian process regression (GPR) to fit a probabilistic model from which replicates may then be drawn. GPR is a Bayesian non-linear regression method. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian (normal) distribution. A GP is defined by a mean function and a covariance function, which specify the mean vectors and covariance matrices for each finite collection of the random variables.<sup id="cite_ref-kirk2009_27-0" class="reference"><a href="#cite_note-kirk2009-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p><b>Regression model:</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)=f(x)+\varepsilon ,\ \ \varepsilon \sim {\mathcal {N}}(0,\sigma ^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>ε</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)=f(x)+\varepsilon ,\ \ \varepsilon \sim {\mathcal {N}}(0,\sigma ^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b07a882e41b5c38fdc629ed2d7153a2da4e674b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.424ex; height:3.176ex;" alt="{\displaystyle y(x)=f(x)+\varepsilon ,\ \ \varepsilon \sim {\mathcal {N}}(0,\sigma ^{2}),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> is a noise term.</dd></dl> <p><b>Gaussian process prior:</b> </p><p>For any finite collection of variables, <i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>, the function outputs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1}),\ldots ,f(x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1}),\ldots ,f(x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b228be7b4ef4feeca30ab70b4c4302ffa3c28b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.286ex; height:2.843ex;" alt="{\displaystyle f(x_{1}),\ldots ,f(x_{n})}"></span> are jointly distributed according to a multivariate Gaussian with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=[m(x_{1}),\ldots ,m(x_{n})]^{\intercal }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=[m(x_{1}),\ldots ,m(x_{n})]^{\intercal }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac3abe6a55b4d0a9df7db10e0c36a58453121a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.388ex; height:2.843ex;" alt="{\displaystyle m=[m(x_{1}),\ldots ,m(x_{n})]^{\intercal }}"></span> and covariance matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K)_{ij}=k(x_{i},x_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K)_{ij}=k(x_{i},x_{j}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7012ca375a075174ffaf176d14e93bd8907126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.521ex; height:3.009ex;" alt="{\displaystyle (K)_{ij}=k(x_{i},x_{j}).}"></span> </p><p>Assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\sim {\mathcal {GP}}(m,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\sim {\mathcal {GP}}(m,k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4bd6ff78b4dab4b71084b7d215658c8c504f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.354ex; height:2.843ex;" alt="{\displaystyle f(x)\sim {\mathcal {GP}}(m,k).}"></span> Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)\sim {\mathcal {GP}}(m,l)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)\sim {\mathcal {GP}}(m,l)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46b940b808ece9d7e50996207a286fcff60128e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.066ex; height:2.843ex;" alt="{\displaystyle y(x)\sim {\mathcal {GP}}(m,l)}"></span>, </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(x_{i},x_{j})=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l(x_{i},x_{j})=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d539ecf8aa6450db3a103060d502c18a18ebb74a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.913ex; height:3.343ex;" alt="{\displaystyle l(x_{i},x_{j})=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j})}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x_{i},x_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x_{i},x_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66de024d3ad7d1e61a6caed9e2606735a6ba4b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.261ex; height:3.009ex;" alt="{\displaystyle \delta (x_{i},x_{j})}"></span> is the standard Kronecker delta function.<sup id="cite_ref-kirk2009_27-1" class="reference"><a href="#cite_note-kirk2009-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p><b>Gaussian process posterior:</b> </p><p>According to GP prior, we can get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [y(x_{1}),\ldots ,y(x_{r})]\sim {\mathcal {N}}(m_{0},K_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [y(x_{1}),\ldots ,y(x_{r})]\sim {\mathcal {N}}(m_{0},K_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44b791e07518721a4bf8b6b0c3df9a90e24e47e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.427ex; height:3.009ex;" alt="{\displaystyle [y(x_{1}),\ldots ,y(x_{r})]\sim {\mathcal {N}}(m_{0},K_{0})}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}=[m(x_{1}),\ldots ,m(x_{r})]^{\intercal }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}=[m(x_{1}),\ldots ,m(x_{r})]^{\intercal }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d40a77e82bd278ecea9d13421f38a71ff88bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.198ex; height:2.843ex;" alt="{\displaystyle m_{0}=[m(x_{1}),\ldots ,m(x_{r})]^{\intercal }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{0})_{ij}=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{0})_{ij}=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53dbff482a01357cc1874962b4fc0178bf31a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.968ex; height:3.343ex;" alt="{\displaystyle (K_{0})_{ij}=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j}).}"></span> </p><p>Let x<sub>1</sub><sup>*</sup>,...,x<sub>s</sub><sup>*</sup> be another finite collection of variables, it's obvious that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [y(x_{1}),\ldots ,y(x_{r}),f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\sim {\mathcal {N}}({\binom {m_{0}}{m_{*}}}{\begin{pmatrix}K_{0}&K_{*}\\K_{*}^{\intercal }&K_{**}\end{pmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [y(x_{1}),\ldots ,y(x_{r}),f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\sim {\mathcal {N}}({\binom {m_{0}}{m_{*}}}{\begin{pmatrix}K_{0}&K_{*}\\K_{*}^{\intercal }&K_{**}\end{pmatrix}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0664dc735f8ff5cc3b0f33671b1d0a2bdbb163db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.674ex; height:6.176ex;" alt="{\displaystyle [y(x_{1}),\ldots ,y(x_{r}),f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\sim {\mathcal {N}}({\binom {m_{0}}{m_{*}}}{\begin{pmatrix}K_{0}&K_{*}\\K_{*}^{\intercal }&K_{**}\end{pmatrix}})}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{*}=[m(x_{1}^{*}),\ldots ,m(x_{s}^{*})]^{\intercal }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{*}=[m(x_{1}^{*}),\ldots ,m(x_{s}^{*})]^{\intercal }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24edcf8081b605823994e85e6c746c87d75a911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.278ex; height:3.009ex;" alt="{\displaystyle m_{*}=[m(x_{1}^{*}),\ldots ,m(x_{s}^{*})]^{\intercal }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{**})_{ij}=k(x_{i}^{*},x_{j}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{**})_{ij}=k(x_{i}^{*},x_{j}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e003a02e0bde931e87c037fecbe48253bbe33847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.056ex; height:3.343ex;" alt="{\displaystyle (K_{**})_{ij}=k(x_{i}^{*},x_{j}^{*})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{*})_{ij}=k(x_{i},x_{j}^{*}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{*})_{ij}=k(x_{i},x_{j}^{*}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b7596f9b64df072d2dfaf7e4d0b93c4a12f425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.627ex; height:3.343ex;" alt="{\displaystyle (K_{*})_{ij}=k(x_{i},x_{j}^{*}).}"></span> </p><p>According to the equations above, the outputs <i>y</i> are also jointly distributed according to a multivariate Gaussian. Thus, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\mid ([y(x)]^{\intercal }=y)\sim {\mathcal {N}}(m_{\text{post}},K_{\text{post}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> <mo>∣</mo> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>post</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>post</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\mid ([y(x)]^{\intercal }=y)\sim {\mathcal {N}}(m_{\text{post}},K_{\text{post}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2dac075f685480962738fefb076bdf3fee90a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.685ex; height:3.176ex;" alt="{\displaystyle [f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\mid ([y(x)]^{\intercal }=y)\sim {\mathcal {N}}(m_{\text{post}},K_{\text{post}}),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=[y_{1},...,y_{r}]^{\intercal }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=[y_{1},...,y_{r}]^{\intercal }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7e5927b40b3dc44746d2f644d0534278b268da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.17ex; height:2.843ex;" alt="{\displaystyle y=[y_{1},...,y_{r}]^{\intercal }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{post}}=m_{*}+K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}(y-m_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>post</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>+</mo> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{post}}=m_{*}+K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}(y-m_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed97896015bc522fa357c002f7c3eb8ebd24be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.293ex; height:3.343ex;" alt="{\displaystyle m_{\text{post}}=m_{*}+K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}(y-m_{0})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\text{post}}=K_{**}-K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}K_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>post</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msub> <mo>−</mo> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊺</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\text{post}}=K_{**}-K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}K_{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e45e525cc7e052d9224602afcee1139078425b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.108ex; height:3.343ex;" alt="{\displaystyle K_{\text{post}}=K_{**}-K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}K_{*}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540b248dfa84cd32dc16402b1cade5866cd6fd2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.997ex; height:2.509ex;" alt="{\displaystyle I_{r}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\times r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>×</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\times r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cafa6704e1fd8023c39b942b2158f07f9bf1fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.938ex; height:1.676ex;" alt="{\displaystyle r\times r}"></span> identity matrix.<sup id="cite_ref-kirk2009_27-2" class="reference"><a href="#cite_note-kirk2009-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Wild_bootstrap">Wild bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=16" title="Edit section: Wild bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The wild bootstrap, proposed originally by Wu (1986),<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> is suited when the model exhibits <a href="/wiki/Heteroskedasticity" class="mw-redirect" title="Heteroskedasticity">heteroskedasticity</a>. The idea is, as the residual bootstrap, to leave the regressors at their sample value, but to resample the response variable based on the residuals values. That is, for each replicate, one computes a new <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> based on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{i}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ε</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{i}v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e904c8f07e7123be2fdd0e7784a6f802edb73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.693ex; height:3.009ex;" alt="{\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{i}v_{i}}"></span></dd></dl> <p>so the residuals are randomly multiplied by a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> with mean 0 and variance 1. For most distributions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> (but not Mammen's), this method assumes that the 'true' residual distribution is symmetric and can offer advantages over simple residual sampling for smaller sample sizes. Different forms are used for the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span>, such as </p> <dl><dd><ul><li>The <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a></li></ul></dd></dl> <dl><dd><ul><li>A distribution suggested by Mammen (1993).<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}={\begin{cases}-({\sqrt {5}}-1)/2&{\text{with probability }}({\sqrt {5}}+1)/(2{\sqrt {5}}),\\({\sqrt {5}}+1)/2&{\text{with probability }}({\sqrt {5}}-1)/(2{\sqrt {5}})\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}={\begin{cases}-({\sqrt {5}}-1)/2&{\text{with probability }}({\sqrt {5}}+1)/(2{\sqrt {5}}),\\({\sqrt {5}}+1)/2&{\text{with probability }}({\sqrt {5}}-1)/(2{\sqrt {5}})\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5a396a1a7562714fd895652225720f9017f041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.546ex; height:6.176ex;" alt="{\displaystyle v_{i}={\begin{cases}-({\sqrt {5}}-1)/2&{\text{with probability }}({\sqrt {5}}+1)/(2{\sqrt {5}}),\\({\sqrt {5}}+1)/2&{\text{with probability }}({\sqrt {5}}-1)/(2{\sqrt {5}})\end{cases}}}"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd>Approximately, Mammen's distribution is:</dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}={\begin{cases}-0.6180\quad {\text{(with a 0 in the units' place)}}&{\text{with probability }}0.7236,\\+1.6180\quad {\text{(with a 1 in the units' place)}}&{\text{with probability }}0.2764.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>0.6180</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(with a 0 in the units' place)</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mn>0.7236</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>1.6180</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(with a 1 in the units' place)</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mn>0.2764.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}={\begin{cases}-0.6180\quad {\text{(with a 0 in the units' place)}}&{\text{with probability }}0.7236,\\+1.6180\quad {\text{(with a 1 in the units' place)}}&{\text{with probability }}0.2764.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ec8bb68c6326c977b5c1b7d8d62d8a21a17683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.584ex; height:6.176ex;" alt="{\displaystyle v_{i}={\begin{cases}-0.6180\quad {\text{(with a 0 in the units' place)}}&{\text{with probability }}0.7236,\\+1.6180\quad {\text{(with a 1 in the units' place)}}&{\text{with probability }}0.2764.\end{cases}}}"></span></dd></dl></dd></dl></dd></dl> <dl><dd><ul><li>Or the simpler distribution, linked to the <a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher distribution</a>:</li></ul></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}={\begin{cases}-1&{\text{with probability }}1/2,\\+1&{\text{with probability }}1/2.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with probability </mtext> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}={\begin{cases}-1&{\text{with probability }}1/2,\\+1&{\text{with probability }}1/2.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1d0dd07f4e62b9d1ba8bf20baa58472bd9430c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.818ex; height:6.176ex;" alt="{\displaystyle v_{i}={\begin{cases}-1&{\text{with probability }}1/2,\\+1&{\text{with probability }}1/2.\end{cases}}}"></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Block_bootstrap">Block bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=17" title="Edit section: Block bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The block bootstrap is used when the data, or the errors in a model, are correlated. In this case, a simple case or residual resampling will fail, as it is not able to replicate the correlation in the data. The block bootstrap tries to replicate the correlation by resampling inside blocks of data (see <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking (statistics)</a>). The block bootstrap has been used mainly with data correlated in time (i.e. time series) but can also be used with data correlated in space, or among groups (so-called cluster data). </p> <div class="mw-heading mw-heading4"><h4 id="Time_series:_Simple_block_bootstrap">Time series: Simple block bootstrap</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=18" title="Edit section: Time series: Simple block bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the (simple) block bootstrap, the variable of interest is split into non-overlapping blocks. </p> <div class="mw-heading mw-heading4"><h4 id="Time_series:_Moving_block_bootstrap">Time series: Moving block bootstrap</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=19" title="Edit section: Time series: Moving block bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the moving block bootstrap, introduced by Künsch (1989),<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> data is split into <i>n</i> − <i>b</i> + 1 overlapping blocks of length <i>b</i>: Observation 1 to b will be block 1, observation 2 to <i>b</i> + 1 will be block 2, etc. Then from these <i>n</i> − <i>b</i> + 1 blocks, <i>n</i>/<i>b</i> blocks will be drawn at random with replacement. Then aligning these n/b blocks in the order they were picked, will give the bootstrap observations. </p><p>This bootstrap works with dependent data, however, the bootstrapped observations will not be stationary anymore by construction. But, it was shown that varying randomly the block length can avoid this problem.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This method is known as the <i>stationary bootstrap.</i> Other related modifications of the moving block bootstrap are the <i>Markovian bootstrap</i> and a stationary bootstrap method that matches subsequent blocks based on standard deviation matching. </p> <div class="mw-heading mw-heading4"><h4 id="Time_series:_Maximum_entropy_bootstrap">Time series: Maximum entropy bootstrap</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=20" title="Edit section: Time series: Maximum entropy bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vinod (2006),<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> presents a method that bootstraps time series data using maximum entropy principles satisfying the Ergodic theorem with mean-preserving and mass-preserving constraints. There is an R package, <b>meboot</b>,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> that utilizes the method, which has applications in econometrics and computer science. </p> <div class="mw-heading mw-heading4"><h4 id="Cluster_data:_block_bootstrap">Cluster data: block bootstrap</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=21" title="Edit section: Cluster data: block bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cluster data describes data where many observations per unit are observed. This could be observing many firms in many states or observing students in many classes. In such cases, the correlation structure is simplified, and one does usually make the assumption that data is correlated within a group/cluster, but independent between groups/clusters. The structure of the block bootstrap is easily obtained (where the block just corresponds to the group), and usually only the groups are resampled, while the observations within the groups are left unchanged. <a href="/wiki/A._Colin_Cameron" title="A. Colin Cameron">Cameron</a> et al. (2008) discusses this for clustered errors in linear regression.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Methods_for_improving_computational_efficiency">Methods for improving computational efficiency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=22" title="Edit section: Methods for improving computational efficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap is a powerful technique although may require substantial computing resources in both time and memory. Some techniques have been developed to reduce this burden. They can generally be combined with many of the different types of Bootstrap schemes and various choices of statistics. </p> <div class="mw-heading mw-heading3"><h3 id="Parallel_processing">Parallel processing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=23" title="Edit section: Parallel processing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most bootstrap methods are <a href="/wiki/Embarrassingly_parallel" title="Embarrassingly parallel">embarrassingly parallel</a> algorithms. That is, the statistic of interest for each bootstrap sample does not depend on other bootstrap samples. Such computations can therefore be performed on separate <a href="/wiki/CPU" class="mw-redirect" title="CPU">CPUs</a> or compute nodes with the results from the separate nodes eventually aggregated for final analysis. </p> <div class="mw-heading mw-heading3"><h3 id="Poisson_bootstrap">Poisson bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=24" title="Edit section: Poisson bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The nonparametric bootstrap samples items from a list of size n with counts drawn from a <a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial distribution</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> denotes the number times element i is included in a given bootstrap sample, then each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> is distributed as a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> with n trials and mean 1, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> is not independent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa98874e6beb16373e8d0e056ba550cf653676a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.103ex; height:2.843ex;" alt="{\displaystyle W_{j}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}"></span>. </p><p>The Poisson bootstrap instead draws samples assuming all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span>'s are independently and identically distributed as Poisson variables with mean 1. The rationale is that the limit of the binomial distribution is Poisson: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\operatorname {Binomial} (n,1/n)=\operatorname {Poisson} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>Binomial</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Poisson</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\operatorname {Binomial} (n,1/n)=\operatorname {Poisson} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30f595c6ab361e94551b74383035ae8e2379c64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.506ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }\operatorname {Binomial} (n,1/n)=\operatorname {Poisson} (1)}"></span></dd></dl> <p>The Poisson bootstrap had been proposed by Hanley and MacGibbon as potentially useful for non-statisticians using software like <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a> and <a href="/wiki/SPSS" title="SPSS">SPSS</a>, which lacked the bootstrap packages of <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> and <a href="/wiki/S-Plus" class="mw-redirect" title="S-Plus">S-Plus</a> programming languages.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> The same authors report that for large enough n, the results are relatively similar to the nonparametric bootstrap estimates but go on to note the Poisson bootstrap has seen minimal use in applications. </p><p>Another proposed advantage of the Poisson bootstrap is the independence of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> makes the method easier to apply for large datasets that must be processed as streams.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>A way to improve on the Poisson bootstrap, termed "sequential bootstrap", is by taking the first samples so that the proportion of unique values is ≈0.632 of the original sample size n. This provides a distribution with main empirical characteristics being within a distance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{3/4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{3/4})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e6fc10daa1bb59b4b3c7b23f419f37062213a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.676ex; height:3.343ex;" alt="{\displaystyle O(n^{3/4})}"></span>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> Empirical investigation has shown this method can yield good results.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> This is related to the reduced bootstrap method.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Bag_of_Little_Bootstraps">Bag of Little Bootstraps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=25" title="Edit section: Bag of Little Bootstraps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For massive data sets, it is often computationally prohibitive to hold all the sample data in memory and resample from the sample data. The Bag of Little Bootstraps (BLB)<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> provides a method of pre-aggregating data before bootstrapping to reduce computational constraints. This works by partitioning the data set into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> equal-sized buckets and aggregating the data within each bucket. This pre-aggregated data set becomes the new sample data over which to draw samples with replacement. This method is similar to the Block Bootstrap, but the motivations and definitions of the blocks are very different. Under certain assumptions, the sample distribution should approximate the full bootstrapped scenario. One constraint is the number of buckets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=n^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=n^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd55979bc525156aa597135477f70684deaf2e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.616ex; height:2.343ex;" alt="{\displaystyle b=n^{\gamma }}"></span>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \in [0.5,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \in [0.5,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6affd7567b75316994441dd287a68607d2c844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.565ex; height:2.843ex;" alt="{\displaystyle \gamma \in [0.5,1]}"></span> and the authors recommend usage of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=n^{0.7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.7</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=n^{0.7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a42af04a28cd4fe765654aae8d3a9e35f15484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.824ex; height:2.676ex;" alt="{\displaystyle b=n^{0.7}}"></span> as a general solution. </p> <div class="mw-heading mw-heading2"><h2 id="Choice_of_statistic">Choice of statistic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=26" title="Edit section: Choice of statistic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap distribution of a point estimator of a population parameter has been used to produce a bootstrapped <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for the parameter's true value if the parameter can be written as a <a href="/wiki/U_statistic#statistical_functional" class="mw-redirect" title="U statistic">function of the population's distribution</a>. </p><p><a href="/wiki/Population_parameter" class="mw-redirect" title="Population parameter">Population parameters</a> are estimated with many <a href="/wiki/Point_estimator" class="mw-redirect" title="Point estimator">point estimators</a>. Popular families of point-estimators include <a href="/wiki/UMVU" class="mw-redirect" title="UMVU">mean-unbiased minimum-variance estimators</a>, <a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">median-unbiased estimators</a>, <a href="/wiki/Bayesian_estimator" class="mw-redirect" title="Bayesian estimator">Bayesian estimators</a> (for example, the <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distribution</a>'s <a href="/wiki/Bayesian_estimator#Posterior_mode" class="mw-redirect" title="Bayesian estimator">mode</a>, <a href="/wiki/Bayesian_estimator#Posterior_median" class="mw-redirect" title="Bayesian estimator">median</a>, <a href="/wiki/Bayesian_estimator#Posterior_mean" class="mw-redirect" title="Bayesian estimator">mean</a>), and <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum-likelihood estimators</a>. </p><p>A Bayesian point estimator and a maximum-likelihood estimator have good performance when the sample size is infinite, according to <a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">asymptotic theory</a>. For practical problems with finite samples, other estimators may be preferable. Asymptotic theory suggests techniques that often improve the performance of bootstrapped estimators; the bootstrapping of a maximum-likelihood estimator may often be improved using transformations related to <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">pivotal quantities</a>.<sup id="cite_ref-BMA_41-0" class="reference"><a href="#cite_note-BMA-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Deriving_confidence_intervals_from_the_bootstrap_distribution">Deriving confidence intervals from the bootstrap distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=27" title="Edit section: Deriving confidence intervals from the bootstrap distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap distribution of a parameter-estimator is often used to calculate <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> for its population-parameter.<sup id="cite_ref-et1993_2-3" class="reference"><a href="#cite_note-et1993-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A variety of methods for constructing the confidence intervals have been proposed, although there is disagreement which method is the best. </p> <div class="mw-heading mw-heading3"><h3 id="Desirable_properties">Desirable properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=28" title="Edit section: Desirable properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The survey of bootstrap confidence interval methods of DiCiccio and Efron and consequent discussion lists several desired properties of confidence intervals, which generally are not all simultaneously met. </p> <ul><li><b>Transformation invariant</b> - the confidence intervals from bootstrapping transformed data (e.g., by taking the logarithm) would ideally be the same as transforming the confidence intervals from bootstrapping the untransformed data.</li> <li>Confidence intervals should be <b>valid</b> or <b>consistent</b>, i.e., the probability a parameter is in a confidence interval with nominal level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }"></span> should be equal to or at least converge in probability to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }"></span>. The latter criteria is both refined and expanded using the framework of Hall.<sup id="cite_ref-Hall1988_42-0" class="reference"><a href="#cite_note-Hall1988-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> The refinements are to distinguish between methods based on how fast the true coverage probability approaches the nominal value, where a method is (using DiCiccio and Efron's terminology) <b>first-order accurate</b> if the error term in the approximation is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1/{\sqrt {n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1/{\sqrt {n}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61ed26c4b4b0c3c3f29bf4d3c6128bfe26510f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.238ex; height:3.009ex;" alt="{\displaystyle O(1/{\sqrt {n}})}"></span> and <b>second-order accurate</b> if the error term is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc17ce8914f0bbdc878918eab8c2fb25163ea66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.31ex; height:3.176ex;" alt="{\displaystyle O(n^{-1})}"></span>. In addition, methods are distinguished by the speed with which the estimated bootstrap critical point converges to the true (unknown) point, and a method is <b>second-order correct</b> when this rate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{p}(n^{-3/2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{p}(n^{-3/2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23845c17b4c53312c2f9e431e00bfc6096b9d218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.013ex; height:3.509ex;" alt="{\displaystyle O_{p}(n^{-3/2})}"></span>.</li> <li>Gleser in the discussion of the paper argues that a limitation of the asymptotic descriptions in the previous bullet is that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82fd5fbde213ba5a358a9b5f47775bc7e716137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle O(\cdot )}"></span> terms are not necessarily uniform in the parameters or true distribution.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Bias,_asymmetry,_and_confidence_intervals"><span id="Bias.2C_asymmetry.2C_and_confidence_intervals"></span>Bias, asymmetry, and confidence intervals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=29" title="Edit section: Bias, asymmetry, and confidence intervals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Bias</b>: The bootstrap distribution and the sample may disagree systematically, in which case <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a> may occur. <dl><dd>If the bootstrap distribution of an estimator is symmetric, then percentile confidence-interval are often used; such intervals are appropriate especially for median-unbiased estimators of minimum risk (with respect to an <a href="/wiki/Absolute_value" title="Absolute value">absolute</a> <a href="/wiki/Loss_function" title="Loss function">loss function</a>). Bias in the bootstrap distribution will lead to bias in the confidence interval.</dd> <dd>Otherwise, if the bootstrap distribution is non-symmetric, then percentile confidence intervals are often inappropriate.</dd></dl></li></ul> <div class="mw-heading mw-heading3"><h3 id="Methods_for_bootstrap_confidence_intervals">Methods for bootstrap confidence intervals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=30" title="Edit section: Methods for bootstrap confidence intervals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several methods for constructing confidence intervals from the bootstrap distribution of a <a href="/wiki/Real_number" title="Real number">real</a> parameter: </p> <ul><li><b>Basic bootstrap</b>,<sup id="cite_ref-BMA_41-1" class="reference"><a href="#cite_note-BMA-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> also known as the <b>Reverse Percentile Interval</b>.<sup id="cite_ref-hesterberg2014teachers_43-0" class="reference"><a href="#cite_note-hesterberg2014teachers-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> The basic bootstrap is a simple scheme to construct the confidence interval: one simply takes the empirical <a href="/wiki/Quantile" title="Quantile">quantiles</a> from the bootstrap distribution of the parameter (see Davison and Hinkley 1997, equ. 5.6 p. 194):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2{\widehat {\theta \,}}-\theta _{(1-\alpha /2)}^{*},2{\widehat {\theta \,}}-\theta _{(\alpha /2)}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2{\widehat {\theta \,}}-\theta _{(1-\alpha /2)}^{*},2{\widehat {\theta \,}}-\theta _{(\alpha /2)}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5de30ed78f8761add2201d4c6cd43ce0473f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:26.5ex; height:4.176ex;" alt="{\displaystyle (2{\widehat {\theta \,}}-\theta _{(1-\alpha /2)}^{*},2{\widehat {\theta \,}}-\theta _{(\alpha /2)}^{*})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{(1-\alpha /2)}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{(1-\alpha /2)}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c1cc85eb03f2d60f647ce0386706888604cdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.398ex; height:3.343ex;" alt="{\displaystyle \theta _{(1-\alpha /2)}^{*}}"></span> denotes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d89cd111892baef3eb29d9b8943859a18ed4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.815ex; height:2.843ex;" alt="{\displaystyle 1-\alpha /2}"></span> <a href="/wiki/Percentile" title="Percentile">percentile</a> of the bootstrapped coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c77b69ba747144843a16bb2e053e9fcd8583735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle \theta ^{*}}"></span>.</dd></dl></dd></dl> <ul><li><b>Percentile bootstrap</b>. The percentile bootstrap proceeds in a similar way to the basic bootstrap, using percentiles of the bootstrap distribution, but with a different formula (note the inversion of the left and right quantiles):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta _{(\alpha /2)}^{*},\theta _{(1-\alpha /2)}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta _{(\alpha /2)}^{*},\theta _{(1-\alpha /2)}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7ea29f8127cdc65a4fc89e44801ef1b0ba4df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:15.54ex; height:3.509ex;" alt="{\displaystyle (\theta _{(\alpha /2)}^{*},\theta _{(1-\alpha /2)}^{*})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{(1-\alpha /2)}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{(1-\alpha /2)}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c1cc85eb03f2d60f647ce0386706888604cdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.398ex; height:3.343ex;" alt="{\displaystyle \theta _{(1-\alpha /2)}^{*}}"></span> denotes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d89cd111892baef3eb29d9b8943859a18ed4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.815ex; height:2.843ex;" alt="{\displaystyle 1-\alpha /2}"></span> <a href="/wiki/Percentile" title="Percentile">percentile</a> of the bootstrapped coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c77b69ba747144843a16bb2e053e9fcd8583735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle \theta ^{*}}"></span>.</dd></dl></dd> <dd>See Davison and Hinkley (1997, equ. 5.18 p. 203) and Efron and Tibshirani (1993, equ 13.5 p. 171).</dd> <dd>This method can be applied to any statistic. It will work well in cases where the bootstrap distribution is symmetrical and centered on the observed statistic<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> and where the sample statistic is median-unbiased and has maximum concentration (or minimum risk with respect to an absolute value loss function). When working with small sample sizes (i.e., less than 50), the basic / reversed percentile and percentile confidence intervals for (for example) the <a href="/wiki/Variance" title="Variance">variance</a> statistic will be too narrow. So that with a sample of 20 points, 90% confidence interval will include the true variance only 78% of the time.<sup id="cite_ref-DAEE_45-0" class="reference"><a href="#cite_note-DAEE-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> The basic / reverse percentile confidence intervals are easier to justify mathematically<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hesterberg2014teachers_43-1" class="reference"><a href="#cite_note-hesterberg2014teachers-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> but they are less accurate in general than percentile confidence intervals, and some authors discourage their use.<sup id="cite_ref-hesterberg2014teachers_43-2" class="reference"><a href="#cite_note-hesterberg2014teachers-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></dd></dl> <ul><li><b><a href="/wiki/Studentization" title="Studentization">Studentized</a> bootstrap</b>. The studentized bootstrap, also called <i>bootstrap-t</i>, is computed analogously to the standard confidence interval, but replaces the quantiles from the normal or student approximation by the quantiles from the bootstrap distribution of the <a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's t-test</a> (see Davison and Hinkley 1997, equ. 5.7 p. 194 and Efron and Tibshirani 1993 equ 12.22, p. 160):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\widehat {\theta \,}}-t_{(1-\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta },{\widehat {\theta \,}}-t_{(\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mtext>se</mtext> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mtext>se</mtext> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\widehat {\theta \,}}-t_{(1-\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta },{\widehat {\theta \,}}-t_{(\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03925c5a96dca02bac1626bc5b737b4442b32701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:33.686ex; height:4.176ex;" alt="{\displaystyle ({\widehat {\theta \,}}-t_{(1-\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta },{\widehat {\theta \,}}-t_{(\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta })}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{(1-\alpha /2)}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{(1-\alpha /2)}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e66e705aebd7d290be85579d4e4aefe052cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.148ex; height:3.343ex;" alt="{\displaystyle t_{(1-\alpha /2)}^{*}}"></span> denotes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d89cd111892baef3eb29d9b8943859a18ed4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.815ex; height:2.843ex;" alt="{\displaystyle 1-\alpha /2}"></span> <a href="/wiki/Percentile" title="Percentile">percentile</a> of the bootstrapped <a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's t-test</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{*}=({\widehat {\theta \,}}^{*}-{\widehat {\theta \,}})/{\widehat {\text{se}}}_{{\widehat {\theta \,}}^{*}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mtext>se</mtext> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{*}=({\widehat {\theta \,}}^{*}-{\widehat {\theta \,}})/{\widehat {\text{se}}}_{{\widehat {\theta \,}}^{*}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9d6d3a3c680c9b1c35cde5c3b5352385facce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.36ex; height:4.009ex;" alt="{\displaystyle t^{*}=({\widehat {\theta \,}}^{*}-{\widehat {\theta \,}})/{\widehat {\text{se}}}_{{\widehat {\theta \,}}^{*}}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\text{se}}}_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mtext>se</mtext> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\text{se}}}_{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00508a097405b2af70186e4a4a6741fd4f5f6d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.338ex; height:2.676ex;" alt="{\displaystyle {\widehat {\text{se}}}_{\theta }}"></span> is the estimated standard error of the coefficient in the original model.</dd></dl></dd></dl> <dl><dd>The studentized test enjoys optimal properties as the statistic that is bootstrapped is <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">pivotal</a> (i.e. it does not depend on <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">nuisance parameters</a> as the t-test follows asymptotically a N(0,1) distribution), unlike the percentile bootstrap.</dd></dl> <ul><li><b>Bias-corrected bootstrap</b> – adjusts for <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a> in the bootstrap distribution.</li> <li><b>Accelerated bootstrap</b> – The bias-corrected and accelerated (BCa) bootstrap, by Efron (1987),<sup id="cite_ref-BCa_13-1" class="reference"><a href="#cite_note-BCa-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> adjusts for both bias and <a href="/wiki/Skewness" title="Skewness">skewness</a> in the bootstrap distribution. This approach is accurate in a wide variety of settings, has reasonable computation requirements, and produces reasonably narrow intervals.<sup id="cite_ref-BCa_13-2" class="reference"><a href="#cite_note-BCa-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bootstrap_hypothesis_testing">Bootstrap hypothesis testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=31" title="Edit section: Bootstrap hypothesis testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>there are other bootstrap tests.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Bootstrapping_(statistics)" title="Special:EditPage/Bootstrapping (statistics)">improve this article</a> if you can.</span> <span class="date-container"><i>(<span class="date">July 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Efron and Tibshirani<sup id="cite_ref-et1993_2-4" class="reference"><a href="#cite_note-et1993-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> suggest the following algorithm for comparing the means of two independent samples: Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}"></span> be a random sample from distribution F with sample mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></span> and sample variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0340dabe0c0edc91ba97a6169d9defbe3fa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}^{2}}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1},\ldots ,y_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1},\ldots ,y_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5b339e24327c7f983d51cb2edaeba7905548f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.009ex;" alt="{\displaystyle y_{1},\ldots ,y_{m}}"></span> be another, independent random sample from distribution G with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690d699292b74ee1b030d5720a17595baaea5227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{y}^{2}}"></span> </p> <ol><li>Calculate the test statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {{\bar {x}}-{\bar {y}}}{\sqrt {\sigma _{x}^{2}/n+\sigma _{y}^{2}/m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\frac {{\bar {x}}-{\bar {y}}}{\sqrt {\sigma _{x}^{2}/n+\sigma _{y}^{2}/m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622b446879a27dcee8f2c84db1bcc3170f631c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:20.583ex; height:8.009ex;" alt="{\displaystyle t={\frac {{\bar {x}}-{\bar {y}}}{\sqrt {\sigma _{x}^{2}/n+\sigma _{y}^{2}/m}}}}"></span></li> <li>Create two new data sets whose values are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}'=x_{i}-{\bar {x}}+{\bar {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}'=x_{i}-{\bar {x}}+{\bar {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e8423a2fe48965025588b7d0ea35848f099c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.664ex; height:2.843ex;" alt="{\displaystyle x_{i}'=x_{i}-{\bar {x}}+{\bar {z}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}'=y_{i}-{\bar {y}}+{\bar {z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}'=y_{i}-{\bar {y}}+{\bar {z}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c0e0666081937a4d39eb625fe795e2fdb61c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.902ex; height:2.843ex;" alt="{\displaystyle y_{i}'=y_{i}-{\bar {y}}+{\bar {z}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dd0599595d539f7d757ec21da6c6e6ac3ad427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}"></span> is the mean of the combined sample.</li> <li>Draw a random sample (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4aad224eaab545bb11af7a02363e59addaad6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle x_{i}^{*}}"></span>) of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> with replacement from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822cb6714de441e54eeb3dcf5224a203e2fec60f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.129ex; height:2.843ex;" alt="{\displaystyle x_{i}'}"></span> and another random sample (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b358e19039c3610c53a3e38a67e3a4509561b3dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.215ex; height:2.843ex;" alt="{\displaystyle y_{i}^{*}}"></span>) of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> with replacement from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d56c61942921f17e2445e654c5c01f4b5b7f641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.939ex; height:2.843ex;" alt="{\displaystyle y_{i}'}"></span>.</li> <li>Calculate the test statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{*}={\frac {{\bar {x^{*}}}-{\bar {y^{*}}}}{\sqrt {\sigma _{x}^{*2}/n+\sigma _{y}^{*2}/m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{*}={\frac {{\bar {x^{*}}}-{\bar {y^{*}}}}{\sqrt {\sigma _{x}^{*2}/n+\sigma _{y}^{*2}/m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1baa65a7d0ea97a8c85a073fedb96d6d31c2100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.166ex; height:8.343ex;" alt="{\displaystyle t^{*}={\frac {{\bar {x^{*}}}-{\bar {y^{*}}}}{\sqrt {\sigma _{x}^{*2}/n+\sigma _{y}^{*2}/m}}}}"></span></li> <li>Repeat 3 and 4 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> times (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=1000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=1000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0637f50d8c9e5fc305bd8faa48357a28d9463f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.512ex; height:2.176ex;" alt="{\displaystyle B=1000}"></span>) to collect <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> values of the test statistic.</li> <li>Estimate the p-value as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {\sum _{i=1}^{B}I\{t_{i}^{*}\geq t\}}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </munderover> <mi>I</mi> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>≥</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> <mi>B</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {\sum _{i=1}^{B}I\{t_{i}^{*}\geq t\}}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e1ae72c91045cbfd8f3b20fb4d8cd6647597dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:20.263ex; height:6.176ex;" alt="{\displaystyle p={\frac {\sum _{i=1}^{B}I\{t_{i}^{*}\geq t\}}{B}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\{{\text{condition}}\}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>condition</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\{{\text{condition}}\}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd07086fc65485a9f8e6e9e0f11c1d8f8b1827da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.191ex; height:2.843ex;" alt="{\displaystyle I\{{\text{condition}}\}=1}"></span> when <i>condition</i> is true and 0 otherwise.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Example_applications">Example applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=32" title="Edit section: Example applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Smoothed_bootstrap">Smoothed bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=33" title="Edit section: Smoothed bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1878, <a href="/wiki/Simon_Newcomb" title="Simon Newcomb">Simon Newcomb</a> took observations on the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> The data set contains two <a href="/wiki/Outlier" title="Outlier">outliers</a>, which greatly influence the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a>. (The sample mean need not be a <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent estimator</a> for any <a href="/wiki/Population_mean" class="mw-redirect" title="Population mean">population mean</a>, because no mean needs to exist for a <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy-tailed distribution</a>.) A well-defined and <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistic</a> for the central tendency is the sample median, which is consistent and <a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">median-unbiased</a> for the population median. </p><p>The bootstrap distribution for Newcomb's data appears below. We can reduce the discreteness of the bootstrap distribution by adding a small amount of random noise to each bootstrap sample. A conventional choice is to add noise with a standard deviation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma /{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma /{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592dd858beccc6d81b0d6156194e6b40bddc2d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.823ex; height:3.009ex;" alt="{\displaystyle \sigma /{\sqrt {n}}}"></span> for a sample size <i>n</i>; this noise is often drawn from a Student-t distribution with <i>n-1</i> degrees of freedom.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> This results in an approximately-unbiased estimator for the variance of the sample mean.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> This means that samples taken from the bootstrap distribution will have a variance which is, on average, equal to the variance of the total population. </p><p>Histograms of the bootstrap distribution and the smooth bootstrap distribution appear below. The bootstrap distribution of the sample-median has only a small number of values. The smoothed bootstrap distribution has a richer <a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">support</a>. However, note that whether the smoothed or standard bootstrap procedure is favorable is case-by-case and is shown to depend on both the underlying distribution function and on the quantity being estimated.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:MedianHists.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/93/MedianHists.png" decoding="async" width="561" height="439" class="mw-file-element" data-file-width="561" data-file-height="439" /></a></span> </p><p>In this example, the bootstrapped 95% (percentile) confidence-interval for the population median is (26, 28.5), which is close to the interval for (25.98, 28.46) for the smoothed bootstrap. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_approaches_to_inference">Relation to other approaches to inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=34" title="Edit section: Relation to other approaches to inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_other_resampling_methods">Relationship to other resampling methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=35" title="Edit section: Relationship to other resampling methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap is distinguished from: </p> <ul><li>the <a href="/wiki/Jackknife_resampling" title="Jackknife resampling">jackknife</a> procedure, used to estimate biases of sample statistics and to estimate variances, and</li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a>, in which the parameters (e.g., regression weights, factor loadings) that are estimated in one subsample are applied to another subsample.</li></ul> <p>For more details see <a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">resampling</a>. </p><p><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bootstrap aggregating</a> (bagging) is a <a href="/wiki/Meta-algorithm" class="mw-redirect" title="Meta-algorithm">meta-algorithm</a> based on averaging model predictions obtained from models trained on multiple bootstrap samples. </p> <div class="mw-heading mw-heading3"><h3 id="U-statistics">U-statistics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=36" title="Edit section: U-statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/U-statistic" title="U-statistic">U-statistic</a></div> <p>In situations where an obvious statistic can be devised to measure a required characteristic using only a small number, <i>r</i>, of data items, a corresponding statistic based on the entire sample can be formulated. Given an <i>r</i>-sample statistic, one can create an <i>n</i>-sample statistic by something similar to bootstrapping (taking the average of the statistic over all subsamples of size <i>r</i>). This procedure is known to have certain good properties and the result is a <a href="/wiki/U-statistic" title="U-statistic">U-statistic</a>. The <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> and <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> are of this form, for <i>r</i> = 1 and <i>r</i> = 2. </p> <div class="mw-heading mw-heading2"><h2 id="Asymptotic_theory">Asymptotic theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=37" title="Edit section: Asymptotic theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap has under certain conditions desirable <a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">asymptotic properties</a>. The asymptotic properties most often described are weak convergence / consistency of the <a href="/wiki/Stochastic_process#Definitions" title="Stochastic process">sample paths</a> of the bootstrap empirical process and the validity of <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> derived from the bootstrap. This section describes the convergence of the empirical bootstrap. </p> <div class="mw-heading mw-heading3"><h3 id="Stochastic_convergence">Stochastic convergence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=38" title="Edit section: Stochastic convergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This paragraph summarizes more complete descriptions of stochastic convergence in van der Vaart and Wellner<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> and Kosorok.<sup id="cite_ref-kosorok2008_52-0" class="reference"><a href="#cite_note-kosorok2008-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> The bootstrap defines a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>, a collection of random variables indexed by some set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is typically the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>) or a family of functions. Processes of interest are those with bounded sample paths, i.e., sample paths in <a href="/wiki/L-infinity" title="L-infinity">L-infinity</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0968d005c13e9345e5efd53aac20b5a59641402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(T)}"></span>), the set of all <a href="/wiki/Uniformly_bounded" class="mw-redirect" title="Uniformly bounded">uniformly bounded</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. When equipped with the uniform distance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0968d005c13e9345e5efd53aac20b5a59641402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(T)}"></span> is a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, and when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48bc9404ad8c49d4c7b79f7fd971b914c3de5bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.413ex; height:2.176ex;" alt="{\displaystyle T=\mathbb {R} }"></span>, two subspaces of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0968d005c13e9345e5efd53aac20b5a59641402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(T)}"></span> are of particular interest, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e160d783783c799aae07cf78d250747461af0ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.419ex; height:2.843ex;" alt="{\displaystyle C[0,1]}"></span>, the space of all <a href="/wiki/Continuous_functions" class="mw-redirect" title="Continuous functions">continuous functions</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> to the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> [0,1], and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d18229131d1ad9cd7cff586ad63243aa6acaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.577ex; height:2.843ex;" alt="{\displaystyle D[0,1]}"></span>, the space of all <a href="/wiki/Cadlag_functions" class="mw-redirect" title="Cadlag functions">cadlag functions</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> to [0,1]. This is because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e160d783783c799aae07cf78d250747461af0ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.419ex; height:2.843ex;" alt="{\displaystyle C[0,1]}"></span> contains the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">distribution functions</a> for all continuous random variables, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d18229131d1ad9cd7cff586ad63243aa6acaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.577ex; height:2.843ex;" alt="{\displaystyle D[0,1]}"></span> contains the distribution functions for all random variables. Statements about the consistency of the bootstrap are statements about the convergence of the sample paths of the bootstrap process as <a href="/wiki/Random_element" title="Random element">random elements</a> of the metric space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0968d005c13e9345e5efd53aac20b5a59641402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(T)}"></span> or some <a href="/wiki/Metric_space#Definition_and_illustrations" title="Metric space">subspace</a> thereof, especially <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e160d783783c799aae07cf78d250747461af0ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.419ex; height:2.843ex;" alt="{\displaystyle C[0,1]}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d18229131d1ad9cd7cff586ad63243aa6acaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.577ex; height:2.843ex;" alt="{\displaystyle D[0,1]}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=39" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Horowitz in a recent review<sup id="cite_ref-Horowitz2019_1-2" class="reference"><a href="#cite_note-Horowitz2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> defines <b>consistency</b> as: the bootstrap estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}(\cdot ,F_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}(\cdot ,F_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7dc54b980b6b6c28879bd10f634658b2517d8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.248ex; height:2.843ex;" alt="{\displaystyle G_{n}(\cdot ,F_{n})}"></span> is consistent [for a statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}"></span>] if, for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f58df5f4307605e8fa07ae29d6262393b3b0c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{0}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{\tau }|G_{n}(\tau ,F_{n})-G_{\infty }(\tau ,F_{0})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{\tau }|G_{n}(\tau ,F_{n})-G_{\infty }(\tau ,F_{0})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44505124b982c16ab7315237033f4fd857679078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.122ex; height:4.343ex;" alt="{\displaystyle \sup _{\tau }|G_{n}(\tau ,F_{n})-G_{\infty }(\tau ,F_{0})|}"></span> <a href="/wiki/Convergence_of_random_variables#Convergence_in_probability" title="Convergence of random variables">converges in probability</a> to 0 as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> is the distribution of the statistic of interest in the original sample, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f58df5f4307605e8fa07ae29d6262393b3b0c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{0}}"></span> is the true but unknown distribution of the statistic, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\infty }(\tau ,F_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\infty }(\tau ,F_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c97647d08dcafdf01096d7c4bd3069874778fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.296ex; height:2.843ex;" alt="{\displaystyle G_{\infty }(\tau ,F_{0})}"></span> is the asymptotic distribution function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is the indexing variable in the distribution function, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T_{n}\leq \tau )=G_{n}(\tau ,F_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T_{n}\leq \tau )=G_{n}(\tau ,F_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f524d9f7dad6c6ec5b83cb385f5e9b5fa32a8582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.169ex; height:2.843ex;" alt="{\displaystyle P(T_{n}\leq \tau )=G_{n}(\tau ,F_{0})}"></span>. This is sometimes more specifically called <b>consistency relative to the Kolmogorov-Smirnov distance</b>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>Horowitz goes on to recommend using a theorem from Mammen<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> that provides easier to check necessary and sufficient conditions for consistency for statistics of a certain common form. In particular, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{i}:i=1,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{i}:i=1,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9161b20ee3ce2b925135130d15bbfad0e886d8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.622ex; height:2.843ex;" alt="{\displaystyle \{X_{i}:i=1,\ldots ,n\}}"></span> be the random sample. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}={\frac {\sum _{i=1}^{n}g_{n}(X_{i})-t_{n}}{\sigma _{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}={\frac {\sum _{i=1}^{n}g_{n}(X_{i})-t_{n}}{\sigma _{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478de3cca9e6637d2752e6ef68ca0e4adf850c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.011ex; height:6.176ex;" alt="{\displaystyle T_{n}={\frac {\sum _{i=1}^{n}g_{n}(X_{i})-t_{n}}{\sigma _{n}}}}"></span> for a sequence of numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4aeaa7d544d1f21afc8c8993cf5e5625ed8f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.546ex; height:2.009ex;" alt="{\displaystyle \sigma _{n}}"></span>, then the bootstrap estimate of the cumulative distribution function estimates the empirical cumulative distribution function if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}"></span> <a href="/wiki/Convergence_of_random_variables#Convergence_in_distribution" title="Convergence of random variables">converges in distribution</a> to the <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Strong_consistency">Strong consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=40" title="Edit section: Strong consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Convergence in (outer) probability as described above is also called <b>weak consistency</b>. It can also be shown with slightly stronger assumptions, that the bootstrap is <b>strongly consistent</b>, where convergence in (outer) probability is replaced by convergence (outer) almost surely. When only one type of consistency is described, it is typically weak consistency. This is adequate for most statistical applications since it implies confidence bands derived from the bootstrap are asymptotically valid.<sup id="cite_ref-kosorok2008_52-1" class="reference"><a href="#cite_note-kosorok2008-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Showing_consistency_using_the_central_limit_theorem">Showing consistency using the central limit theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=41" title="Edit section: Showing consistency using the central limit theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In simpler cases, it is possible to use the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> directly to show the <a href="/wiki/Consistency_(statistics)" title="Consistency (statistics)">consistency</a> of the bootstrap procedure for estimating the distribution of the sample mean. </p><p>Specifically, let us consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n1},\ldots ,X_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n1},\ldots ,X_{nn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f27a30cd6b8fcf734e2e264e2bde9b8ced6259d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.272ex; height:2.509ex;" alt="{\displaystyle X_{n1},\ldots ,X_{nn}}"></span> independent identically distributed random variables with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [X_{n1}]=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{n1}]=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16616fa7aae5c68ea292668e1be4b09912511b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{n1}]=\mu }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Var}}[X_{n1}]=\sigma ^{2}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Var</mtext> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Var}}[X_{n1}]=\sigma ^{2}<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b3ef8f9245a4ee21895f709c8366fba074e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.981ex; height:3.176ex;" alt="{\displaystyle {\text{Var}}[X_{n1}]=\sigma ^{2}<\infty }"></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}=n^{-1}(X_{n1}+\cdots +X_{nn})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}=n^{-1}(X_{n1}+\cdots +X_{nn})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789d2ff0504f156100016a974bb088a35197f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.331ex; height:3.176ex;" alt="{\displaystyle {\bar {X}}_{n}=n^{-1}(X_{n1}+\cdots +X_{nn})}"></span>. In addition, for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>, conditional on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n1},\ldots ,X_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n1},\ldots ,X_{nn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f27a30cd6b8fcf734e2e264e2bde9b8ced6259d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.272ex; height:2.509ex;" alt="{\displaystyle X_{n1},\ldots ,X_{nn}}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n1}^{*},\ldots ,X_{nn}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n1}^{*},\ldots ,X_{nn}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65feb3ed69d688169f1a4b1086d47aa4499d0fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.272ex; height:2.843ex;" alt="{\displaystyle X_{n1}^{*},\ldots ,X_{nn}^{*}}"></span> be independent random variables with distribution equal to the empirical distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n1},\ldots ,X_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n1},\ldots ,X_{nn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f27a30cd6b8fcf734e2e264e2bde9b8ced6259d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.272ex; height:2.509ex;" alt="{\displaystyle X_{n1},\ldots ,X_{nn}}"></span>. This is the sequence of bootstrap samples. </p><p>Then it can be shown that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{\tau \in \mathbb {R} }\left|P^{*}\left({\frac {{\sqrt {n}}({\bar {X}}_{n}^{*}-{\bar {X}}_{n})}{{\hat {\sigma }}_{n}}}\leq \tau \right)-P\left({\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq \tau \right)\right|\to 0{\text{ in probability as }}n\to \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mi>τ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>≤</mo> <mi>τ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> in probability as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{\tau \in \mathbb {R} }\left|P^{*}\left({\frac {{\sqrt {n}}({\bar {X}}_{n}^{*}-{\bar {X}}_{n})}{{\hat {\sigma }}_{n}}}\leq \tau \right)-P\left({\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq \tau \right)\right|\to 0{\text{ in probability as }}n\to \infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57c01e47a1278235f8cf769ddead0e4d7c5ef45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.295ex; height:7.509ex;" alt="{\displaystyle \sup _{\tau \in \mathbb {R} }\left|P^{*}\left({\frac {{\sqrt {n}}({\bar {X}}_{n}^{*}-{\bar {X}}_{n})}{{\hat {\sigma }}_{n}}}\leq \tau \right)-P\left({\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq \tau \right)\right|\to 0{\text{ in probability as }}n\to \infty ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24635cb75ac6b04bbdb6054de6df7eca458d3ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.876ex; height:2.343ex;" alt="{\displaystyle P^{*}}"></span> represents probability conditional on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n1},\ldots ,X_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n1},\ldots ,X_{nn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f27a30cd6b8fcf734e2e264e2bde9b8ced6259d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.272ex; height:2.509ex;" alt="{\displaystyle X_{n1},\ldots ,X_{nn}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}^{*}=n^{-1}(X_{n1}^{*}+\cdots +X_{nn}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}^{*}=n^{-1}(X_{n1}^{*}+\cdots +X_{nn}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84170e7a4e5945ef834843816770356c37f1c921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.331ex; height:3.343ex;" alt="{\displaystyle {\bar {X}}_{n}^{*}=n^{-1}(X_{n1}^{*}+\cdots +X_{nn}^{*})}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}_{n}^{2}=n^{-1}\sum _{i=1}^{n}(X_{ni}-{\bar {X}}_{n})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}_{n}^{2}=n^{-1}\sum _{i=1}^{n}(X_{ni}-{\bar {X}}_{n})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c22b34d548e61a197f15a77558441e22239afcdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.729ex; height:6.843ex;" alt="{\displaystyle {\hat {\sigma }}_{n}^{2}=n^{-1}\sum _{i=1}^{n}(X_{ni}-{\bar {X}}_{n})^{2}}"></span>. </p><p>To see this, note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{ni}^{*}-{\bar {X}}_{n})/{\sqrt {n}}{\hat {\sigma }}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{ni}^{*}-{\bar {X}}_{n})/{\sqrt {n}}{\hat {\sigma }}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9f616be9295af94e90c375715c8da530f40db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.6ex; height:3.176ex;" alt="{\displaystyle (X_{ni}^{*}-{\bar {X}}_{n})/{\sqrt {n}}{\hat {\sigma }}_{n}}"></span> satisfies the <a href="/wiki/Lindeberg_condition" class="mw-redirect" title="Lindeberg condition">Lindeberg condition</a>, so the CLT holds.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Glivenko%E2%80%93Cantelli_theorem" title="Glivenko–Cantelli theorem">Glivenko–Cantelli theorem</a> provides theoretical background for the bootstrap method. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=42" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">Accuracy and precision</a></li> <li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bootstrap aggregating</a></li> <li><a href="/wiki/Bootstrapping" title="Bootstrapping">Bootstrapping</a></li> <li><a href="/wiki/Empirical_likelihood" title="Empirical likelihood">Empirical likelihood</a></li> <li><a href="/wiki/Imputation_(statistics)" title="Imputation (statistics)">Imputation (statistics)</a></li> <li><a href="/wiki/Reliability_(statistics)" title="Reliability (statistics)">Reliability (statistics)</a></li> <li><a href="/wiki/Reproducibility" title="Reproducibility">Reproducibility</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=43" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Horowitz2019-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Horowitz2019_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Horowitz2019_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Horowitz2019_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHorowitz2019" class="citation journal cs1">Horowitz JL (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-economics-080218-025651">"Bootstrap methods in econometrics"</a>. <i>Annual Review of Economics</i>. <b>11</b>: 193–224. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1809.04016">1809.04016</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-economics-080218-025651">10.1146/annurev-economics-080218-025651</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Economics&rft.atitle=Bootstrap+methods+in+econometrics&rft.volume=11&rft.pages=193-224&rft.date=2019&rft_id=info%3Aarxiv%2F1809.04016&rft_id=info%3Adoi%2F10.1146%2Fannurev-economics-080218-025651&rft.aulast=Horowitz&rft.aufirst=Joel+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1146%252Fannurev-economics-080218-025651&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-et1993-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-et1993_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-et1993_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-et1993_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-et1993_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-et1993_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfronTibshirani1993" class="citation book cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a>, <a href="/wiki/Robert_Tibshirani" title="Robert Tibshirani">Tibshirani R</a> (1993). <i>An Introduction to the Bootstrap</i>. Boca Raton, FL: Chapman & Hall/CRC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-04231-2" title="Special:BookSources/0-412-04231-2"><bdi>0-412-04231-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Bootstrap&rft.place=Boca+Raton%2C+FL&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=1993&rft.isbn=0-412-04231-2&rft.aulast=Efron&rft.aufirst=B.&rft.au=Tibshirani%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://lib.stat.cmu.edu/S/bootstrap.funs">software</a> <a rel="nofollow" class="external text" href="https://archive.today/20120712124533/http://lib.stat.cmu.edu/S/bootstrap.funs">Archived</a> 2012-07-12 at <a href="/wiki/Archive.today" title="Archive.today">archive.today</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron2003" class="citation journal cs1">Efron B (2003). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/statistical-science/volume-18/issue-2/Second-Thoughts-on-the-Bootstrap/10.1214/ss/1063994968.pdf">"Second thoughts on the bootstrap"</a> <span class="cs1-format">(PDF)</span>. <i>Statistical Science</i>. <b>18</b> (2): 135–140. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1063994968">10.1214/ss/1063994968</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=Second+thoughts+on+the+bootstrap&rft.volume=18&rft.issue=2&rft.pages=135-140&rft.date=2003&rft_id=info%3Adoi%2F10.1214%2Fss%2F1063994968&rft.aulast=Efron&rft.aufirst=Bradley&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fstatistical-science%2Fvolume-18%2Fissue-2%2FSecond-Thoughts-on-the-Bootstrap%2F10.1214%2Fss%2F1063994968.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Efron1979-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Efron1979_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Efron1979_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Efron1979_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron,_B.1979" class="citation journal cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron, B.</a> (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">"Bootstrap methods: Another look at the jackknife"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>7</b> (1): 1–26. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">10.1214/aos/1176344552</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Bootstrap+methods%3A+Another+look+at+the+jackknife&rft.volume=7&rft.issue=1&rft.pages=1-26&rft.date=1979&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176344552&rft.au=Efron%2C+B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176344552&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Lehmann E.L. (1992) "Introduction to Neyman and Pearson (1933) On the Problem of the Most Efficient Tests of Statistical Hypotheses". In: Breakthroughs in Statistics, Volume 1, (Eds Kotz, S., Johnson, N.L.), Springer-Verlag. ISBN 0-387-94037-5 (followed by reprinting of the paper).</span> </li> <li id="cite_note-Quenouille1949-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Quenouille1949_7-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuenouille1949" class="citation journal cs1"><a href="/wiki/Maurice_Quenouille" title="Maurice Quenouille">Quenouille MH</a> (1949). "Approximate tests of correlation in time-series". <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" class="mw-redirect" title="Journal of the Royal Statistical Society, Series B">Journal of the Royal Statistical Society, Series B</a></i>. <b>11</b> (1): 68–84. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.2517-6161.1949.tb00023.x">10.1111/j.2517-6161.1949.tb00023.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=Approximate+tests+of+correlation+in+time-series&rft.volume=11&rft.issue=1&rft.pages=68-84&rft.date=1949&rft_id=info%3Adoi%2F10.1111%2Fj.2517-6161.1949.tb00023.x&rft.aulast=Quenouille&rft.aufirst=M.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Tukey1958-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tukey1958_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTukey" class="citation journal cs1"><a href="/wiki/John_Tukey" title="John Tukey">Tukey JW</a>. "Bias and confidence in not-quite large samples". <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>29</b>: 614.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Bias+and+confidence+in+not-quite+large+samples&rft.volume=29&rft.pages=614&rft.aulast=Tukey&rft.aufirst=J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Jaeckel1972-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jaeckel1972_9-0">^</a></b></span> <span class="reference-text">Jaeckel L (1972) The infinitesimal jackknife. Memorandum MM72-1215-11, Bell Lab</span> </li> <li id="cite_note-Bickel1982-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bickel1982_10-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBickelFreedman1981" class="citation journal cs1"><a href="/wiki/Peter_J._Bickel" title="Peter J. Bickel">Bickel PJ</a>, <a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman DA</a> (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345637">"Some asymptotic theory for the bootstrap"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>9</b> (6): 1196–1217. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345637">10.1214/aos/1176345637</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Some+asymptotic+theory+for+the+bootstrap&rft.volume=9&rft.issue=6&rft.pages=1196-1217&rft.date=1981&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176345637&rft.aulast=Bickel&rft.aufirst=Peter+J.&rft.au=Freedman%2C+David+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176345637&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Singh1981-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Singh1981_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh1981" class="citation journal cs1">Singh K (1981). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2240409">"On the asymptotic accuracy of Efron's bootstrap"</a></span>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>9</b> (6): 1187–1195. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345636">10.1214/aos/1176345636</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2240409">2240409</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=On+the+asymptotic+accuracy+of+Efron%27s+bootstrap&rft.volume=9&rft.issue=6&rft.pages=1187-1195&rft.date=1981&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176345636&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2240409%23id-name%3DJSTOR&rft.aulast=Singh&rft.aufirst=Kesar&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2240409&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Rubin1981-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rubin1981_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRubin1981" class="citation journal cs1"><a href="/wiki/Donald_B._Rubin" class="mw-redirect" title="Donald B. Rubin">Rubin DB</a> (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345338">"The Bayesian bootstrap"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>9</b>: 130–134. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345338">10.1214/aos/1176345338</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=The+Bayesian+bootstrap&rft.volume=9&rft.pages=130-134&rft.date=1981&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176345338&rft.aulast=Rubin&rft.aufirst=Donald+B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176345338&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-BCa-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-BCa_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BCa_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-BCa_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron,_B.1987" class="citation journal cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron, B.</a> (1987). "Better Bootstrap Confidence Intervals". <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>. <b>82</b> (397). Journal of the American Statistical Association, Vol. 82, No. 397: 171–185. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2289144">10.2307/2289144</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2289144">2289144</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Better+Bootstrap+Confidence+Intervals&rft.volume=82&rft.issue=397&rft.pages=171-185&rft.date=1987&rft_id=info%3Adoi%2F10.2307%2F2289144&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2289144%23id-name%3DJSTOR&rft.au=Efron%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Diciccio1992-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Diciccio1992_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiCiccioEfron1992" class="citation journal cs1">DiCiccio T, <a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (1992). <a rel="nofollow" class="external text" href="https://academic.oup.com/biomet/article-abstract/79/2/231/225847">"More accurate confidence intervals in exponential families"</a>. <i>Biometrika</i>. <b>79</b> (2): 231–245. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2336835">10.2307/2336835</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-3444">0006-3444</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2336835">2336835</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/5545447518">5545447518</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=More+accurate+confidence+intervals+in+exponential+families&rft.volume=79&rft.issue=2&rft.pages=231-245&rft.date=1992&rft_id=info%3Aoclcnum%2F5545447518&rft.issn=0006-3444&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2336835%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2336835&rft.aulast=DiCiccio&rft.aufirst=Thomas&rft.au=Efron%2C+Bradley&rft_id=https%3A%2F%2Facademic.oup.com%2Fbiomet%2Farticle-abstract%2F79%2F2%2F231%2F225847&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Good, P. (2006) Resampling Methods. 3rd Ed. Birkhauser.</span> </li> <li id="cite_note-ch21_brm-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-ch21_brm_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ch21_brm_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ch21_brm_16-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.sagepub.com/sites/default/files/upm-binaries/21122_Chapter_21.pdf">"21 Bootstrapping Regression Models"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150724033201/http://www.sagepub.com:80/sites/default/files/upm-binaries/21122_Chapter_21.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2015-07-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=21+Bootstrapping+Regression+Models&rft_id=https%3A%2F%2Fwww.sagepub.com%2Fsites%2Fdefault%2Ffiles%2Fupm-binaries%2F21122_Chapter_21.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-DiCiccio1996-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-DiCiccio1996_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiCiccioEfron1996" class="citation journal cs1">DiCiccio TJ, <a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1032280214">"Bootstrap confidence intervals (with Discussion)"</a>. <i>Statistical Science</i>. <b>11</b> (3): 189–228. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1032280214">10.1214/ss/1032280214</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=Bootstrap+confidence+intervals+%28with+Discussion%29&rft.volume=11&rft.issue=3&rft.pages=189-228&rft.date=1996&rft_id=info%3Adoi%2F10.1214%2Fss%2F1032280214&rft.aulast=DiCiccio&rft.aufirst=Thomas+J.&rft.au=Efron%2C+Bradley&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Fss%252F1032280214&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinkley1994" class="citation journal cs1">Hinkley D (1994). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177010387">"[Bootstrap: More than a Stab in the Dark?]: Comment"</a>. <i>Statistical Science</i>. <b>9</b> (3): 400–403. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177010387">10.1214/ss/1177010387</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0883-4237">0883-4237</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=%5BBootstrap%3A+More+than+a+Stab+in+the+Dark%3F%5D%3A+Comment&rft.volume=9&rft.issue=3&rft.pages=400-403&rft.date=1994&rft_id=info%3Adoi%2F10.1214%2Fss%2F1177010387&rft.issn=0883-4237&rft.aulast=Hinkley&rft.aufirst=David&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Fss%252F1177010387&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodhueLewisThompson2012" class="citation journal cs1">Goodhue DL, Lewis W, Thompson W (2012). "Does PLS have advantages for small sample size or non-normal data?". <i><a href="/wiki/MIS_Quarterly" class="mw-redirect" title="MIS Quarterly">MIS Quarterly</a></i>. <b>36</b> (3): 981–1001. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F41703490">10.2307/41703490</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41703490">41703490</a>. <a rel="nofollow" class="external text" href="https://www.misq.org/skin/frontend/default/misq/pdf/appendices/2012/V36I3_Appendices/GoodhueResearchNoteAppendix.pdf">Appendix</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=MIS+Quarterly&rft.atitle=Does+PLS+have+advantages+for+small+sample+size+or+non-normal+data%3F&rft.volume=36&rft.issue=3&rft.pages=981-1001&rft.date=2012&rft_id=info%3Adoi%2F10.2307%2F41703490&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41703490%23id-name%3DJSTOR&rft.aulast=Goodhue&rft.aufirst=Dale+L.&rft.au=Lewis%2C+William&rft.au=Thompson%2C+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Efron, B., Rogosa, D., & Tibshirani, R. (2004). Resampling methods of estimation. In N.J. Smelser, & P.B. Baltes (Eds.). International Encyclopedia of the Social & Behavioral Sciences (pp. 13216–13220). New York, NY: Elsevier.</span> </li> <li id="cite_note-Ader-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ader_21-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Ad%C3%A8r,_H._J." class="mw-redirect" title="Adèr, H. J.">Adèr, H. J.</a>, <a href="/wiki/Gideon_J._Mellenbergh" title="Gideon J. Mellenbergh">Mellenbergh G. J.</a>, & Hand, D. J. (2008). <i>Advising on research methods: A consultant's companion</i>. Huizen, The Netherlands: Johannes van Kessel Publishing. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-79418-01-5" title="Special:BookSources/978-90-79418-01-5">978-90-79418-01-5</a>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAthreya1987" class="citation journal cs1">Athreya KB (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350371">"Bootstrap of the mean in the infinite variance case"</a>. <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a></i>. <b>15</b> (2): 724–731. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350371">10.1214/aos/1176350371</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=Bootstrap+of+the+mean+in+the+infinite+variance+case&rft.volume=15&rft.issue=2&rft.pages=724-731&rft.date=1987&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176350371&rft.aulast=Athreya&rft.aufirst=K.+B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176350371&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190914102512/http://statweb.stanford.edu/~susan/courses/s208/node37.html">"How many different bootstrap samples are there? Statweb.stanford.edu"</a>. Archived from <a rel="nofollow" class="external text" href="http://statweb.stanford.edu/~susan/courses/s208/node37.html">the original</a> on 2019-09-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+many+different+bootstrap+samples+are+there%3F+Statweb.stanford.edu&rft_id=http%3A%2F%2Fstatweb.stanford.edu%2F~susan%2Fcourses%2Fs208%2Fnode37.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Rubin, D. B. (1981). "The Bayesian bootstrap". <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a></i>, 9, 130.</span> </li> <li id="cite_note-wang1995-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-wang1995_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wang1995_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWANG1995" class="citation journal cs1">WANG, SUOJIN (1995). "Optimizing the smoothed bootstrap". <i>Ann. Inst. Statist. Math</i>. <b>47</b>: 65–80. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00773412">10.1007/BF00773412</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122041565">122041565</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Inst.+Statist.+Math.&rft.atitle=Optimizing+the+smoothed+bootstrap&rft.volume=47&rft.pages=65-80&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2FBF00773412&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122041565%23id-name%3DS2CID&rft.aulast=WANG&rft.aufirst=SUOJIN&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekkingKraaikampLopuhaäMeester2005" class="citation book cs1">Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). <i>A modern introduction to probability and statistics : understanding why and how</i>. London: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-896-1" title="Special:BookSources/978-1-85233-896-1"><bdi>978-1-85233-896-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/262680588">262680588</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+introduction+to+probability+and+statistics+%3A+understanding+why+and+how&rft.place=London&rft.pub=Springer&rft.date=2005&rft_id=info%3Aoclcnum%2F262680588&rft.isbn=978-1-85233-896-1&rft.aulast=Dekking&rft.aufirst=Frederik+Michel&rft.au=Kraaikamp%2C+Cornelis&rft.au=Lopuha%C3%A4%2C+Hendrik+Paul&rft.au=Meester%2C+Ludolf+Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-kirk2009-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-kirk2009_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kirk2009_27-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-kirk2009_27-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirk2009" class="citation journal cs1">Kirk, Paul (2009). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2677737">"Gaussian process regression bootstrapping: exploring the effects of uncertainty in time course data"</a>. <i>Bioinformatics</i>. <b>25</b> (10): 1300–1306. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtp139">10.1093/bioinformatics/btp139</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2677737">2677737</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19289448">19289448</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bioinformatics&rft.atitle=Gaussian+process+regression+bootstrapping%3A+exploring+the+effects+of+uncertainty+in+time+course+data&rft.volume=25&rft.issue=10&rft.pages=1300-1306&rft.date=2009&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2677737%23id-name%3DPMC&rft_id=info%3Apmid%2F19289448&rft_id=info%3Adoi%2F10.1093%2Fbioinformatics%2Fbtp139&rft.aulast=Kirk&rft.aufirst=Paul&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2677737&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWu,_C.F.J.1986" class="citation journal cs1">Wu, C.F.J. (1986). <a rel="nofollow" class="external text" href="http://dml.cz/bitstream/handle/10338.dmlcz/135473/Kybernetika_38-2002-4_1.pdf">"Jackknife, bootstrap and other resampling methods in regression analysis (with discussions)"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Statistics</i>. <b>14</b>: 1261–1350. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350142">10.1214/aos/1176350142</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=Jackknife%2C+bootstrap+and+other+resampling+methods+in+regression+analysis+%28with+discussions%29&rft.volume=14&rft.pages=1261-1350&rft.date=1986&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176350142&rft.au=Wu%2C+C.F.J.&rft_id=http%3A%2F%2Fdml.cz%2Fbitstream%2Fhandle%2F10338.dmlcz%2F135473%2FKybernetika_38-2002-4_1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMammen,_E.1993" class="citation journal cs1">Mammen, E. (Mar 1993). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176349025">"Bootstrap and wild bootstrap for high dimensional linear models"</a>. <i>Annals of Statistics</i>. <b>21</b> (1): 255–285. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176349025">10.1214/aos/1176349025</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=Bootstrap+and+wild+bootstrap+for+high+dimensional+linear+models&rft.volume=21&rft.issue=1&rft.pages=255-285&rft.date=1993-03&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176349025&rft.au=Mammen%2C+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176349025&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKünsch1989" class="citation journal cs1">Künsch, H. R. (1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176347265">"The Jackknife and the Bootstrap for General Stationary Observations"</a>. <i>Annals of Statistics</i>. <b>17</b> (3): 1217–1241. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176347265">10.1214/aos/1176347265</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=The+Jackknife+and+the+Bootstrap+for+General+Stationary+Observations&rft.volume=17&rft.issue=3&rft.pages=1217-1241&rft.date=1989&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176347265&rft.aulast=K%C3%BCnsch&rft.aufirst=H.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176347265&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolitisRomano1994" class="citation journal cs1">Politis, D. N.; Romano, J. P. (1994). "The Stationary Bootstrap". <i>Journal of the American Statistical Association</i>. <b>89</b> (428): 1303–1313. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1994.10476870">10.1080/01621459.1994.10476870</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10983%2F25607">10983/25607</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=The+Stationary+Bootstrap&rft.volume=89&rft.issue=428&rft.pages=1303-1313&rft.date=1994&rft_id=info%3Ahdl%2F10983%2F25607&rft_id=info%3Adoi%2F10.1080%2F01621459.1994.10476870&rft.aulast=Politis&rft.aufirst=D.+N.&rft.au=Romano%2C+J.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinod2006" class="citation journal cs1">Vinod, HD (2006). "Maximum entropy ensembles for time series inference in economics". <i>Journal of Asian Economics</i>. <b>17</b> (6): 955–978. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.asieco.2006.09.001">10.1016/j.asieco.2006.09.001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Asian+Economics&rft.atitle=Maximum+entropy+ensembles+for+time+series+inference+in+economics&rft.volume=17&rft.issue=6&rft.pages=955-978&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.asieco.2006.09.001&rft.aulast=Vinod&rft.aufirst=HD&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinodLópez-de-Lacalle2009" class="citation journal cs1">Vinod, Hrishikesh; López-de-Lacalle, Javier (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v029.i05">"Maximum entropy bootstrap for time series: The meboot R package"</a>. <i>Journal of Statistical Software</i>. <b>29</b> (5): 1–19. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v029.i05">10.18637/jss.v029.i05</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Software&rft.atitle=Maximum+entropy+bootstrap+for+time+series%3A+The+meboot+R+package&rft.volume=29&rft.issue=5&rft.pages=1-19&rft.date=2009&rft_id=info%3Adoi%2F10.18637%2Fjss.v029.i05&rft.aulast=Vinod&rft.aufirst=Hrishikesh&rft.au=L%C3%B3pez-de-Lacalle%2C+Javier&rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v029.i05&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCameronGelbachMiller2008" class="citation journal cs1">Cameron, A. C.; Gelbach, J. B.; Miller, D. L. (2008). <a rel="nofollow" class="external text" href="http://www.nber.org/papers/t0344.pdf">"Bootstrap-based improvements for inference with clustered errors"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Review_of_Economics_and_Statistics" class="mw-redirect" title="Review of Economics and Statistics">Review of Economics and Statistics</a></i>. <b>90</b> (3): 414–427. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Frest.90.3.414">10.1162/rest.90.3.414</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Review+of+Economics+and+Statistics&rft.atitle=Bootstrap-based+improvements+for+inference+with+clustered+errors&rft.volume=90&rft.issue=3&rft.pages=414-427&rft.date=2008&rft_id=info%3Adoi%2F10.1162%2Frest.90.3.414&rft.aulast=Cameron&rft.aufirst=A.+C.&rft.au=Gelbach%2C+J.+B.&rft.au=Miller%2C+D.+L.&rft_id=http%3A%2F%2Fwww.nber.org%2Fpapers%2Ft0344.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanleyMacGibbon2006" class="citation journal cs1">Hanley JA, MacGibbon B (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cmpb.2006.04.006">"Creating non-parametric bootstrap samples using Poisson frequencies"</a>. <i>Computer Methods and Programs in Biomedicine</i>. <b>83</b> (1): 57–62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cmpb.2006.04.006">10.1016/j.cmpb.2006.04.006</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16730851">16730851</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Methods+and+Programs+in+Biomedicine&rft.atitle=Creating+non-parametric+bootstrap+samples+using+Poisson+frequencies&rft.volume=83&rft.issue=1&rft.pages=57-62&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.cmpb.2006.04.006&rft_id=info%3Apmid%2F16730851&rft.aulast=Hanley&rft.aufirst=James+A.&rft.au=MacGibbon%2C+Brenda&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.cmpb.2006.04.006&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChamandyMuralidharanNajmiNaidu2012" class="citation web cs1">Chamandy N, Muralidharan O, Najmi A, Naidu S (2012). <a rel="nofollow" class="external text" href="http://www.unofficialgoogledatascience.com/2015/08/an-introduction-to-poisson-bootstrap26.html">"Estimating Uncertainty for Massive Data Streams"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Estimating+Uncertainty+for+Massive+Data+Streams&rft.date=2012&rft.aulast=Chamandy&rft.aufirst=N.&rft.au=Muralidharan%2C+O.&rft.au=Najmi%2C+A.&rft.au=Naidu%2C+S.&rft_id=http%3A%2F%2Fwww.unofficialgoogledatascience.com%2F2015%2F08%2Fan-introduction-to-poisson-bootstrap26.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBabuPathakRao1999" class="citation journal cs1">Babu, G. Jogesh; Pathak, P. K; <a href="/wiki/C._R._Rao" title="C. R. Rao">Rao, C. R.</a> (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1017939146">"Second-order correctness of the Poisson bootstrap"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>27</b> (5): 1666–1683. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1017939146">10.1214/aos/1017939146</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Second-order+correctness+of+the+Poisson+bootstrap&rft.volume=27&rft.issue=5&rft.pages=1666-1683&rft.date=1999&rft_id=info%3Adoi%2F10.1214%2Faos%2F1017939146&rft.aulast=Babu&rft.aufirst=G.+Jogesh&rft.au=Pathak%2C+P.+K&rft.au=Rao%2C+C.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1017939146&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShoemakerPathak2001" class="citation journal cs1">Shoemaker, Owen J.; Pathak, P. K. (2001). "The sequential bootstrap: a comparison with regular bootstrap". <i><a href="/wiki/Communications_in_Statistics_-_Theory_and_Methods" class="mw-redirect" title="Communications in Statistics - Theory and Methods">Communications in Statistics - Theory and Methods</a></i>. <b>30</b> (8–9): 1661–1674. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1081%2FSTA-100105691">10.1081/STA-100105691</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&rft.atitle=The+sequential+bootstrap%3A+a+comparison+with+regular+bootstrap&rft.volume=30&rft.issue=8%E2%80%939&rft.pages=1661-1674&rft.date=2001&rft_id=info%3Adoi%2F10.1081%2FSTA-100105691&rft.aulast=Shoemaker&rft.aufirst=Owen+J.&rft.au=Pathak%2C+P.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiménez-GameroMuñoz-GarcíaPino-Mejías2004" class="citation journal cs1">Jiménez-Gamero, María Dolores; Muñoz-García, Joaquín; Pino-Mejías, Rafael (2004). "Reduced bootstrap for the median". <i><a href="/wiki/Statistica_Sinica" title="Statistica Sinica">Statistica Sinica</a></i>. <b>14</b> (4): 1179–1198. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24307226">24307226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistica+Sinica&rft.atitle=Reduced+bootstrap+for+the+median&rft.volume=14&rft.issue=4&rft.pages=1179-1198&rft.date=2004&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24307226%23id-name%3DJSTOR&rft.aulast=Jim%C3%A9nez-Gamero&rft.aufirst=Mar%C3%ADa+Dolores&rft.au=Mu%C3%B1oz-Garc%C3%ADa%2C+Joaqu%C3%ADn&rft.au=Pino-Mej%C3%ADas%2C+Rafael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinerTalwalkarSarkarJordan2014" class="citation journal cs1">Kleiner, A; Talwalkar, A; Sarkar, P; Jordan, M. I. (2014). "A scalable bootstrap for massive data". <i>Journal of the Royal Statistical Society, Series B (Statistical Methodology)</i>. <b>76</b> (4): 795–816. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.5016">1112.5016</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Frssb.12050">10.1111/rssb.12050</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1369-7412">1369-7412</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3064206">3064206</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B+%28Statistical+Methodology%29&rft.atitle=A+scalable+bootstrap+for+massive+data&rft.volume=76&rft.issue=4&rft.pages=795-816&rft.date=2014&rft_id=info%3Aarxiv%2F1112.5016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3064206%23id-name%3DS2CID&rft.issn=1369-7412&rft_id=info%3Adoi%2F10.1111%2Frssb.12050&rft.aulast=Kleiner&rft.aufirst=A&rft.au=Talwalkar%2C+A&rft.au=Sarkar%2C+P&rft.au=Jordan%2C+M.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-BMA-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-BMA_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BMA_41-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisonHinkley1997" class="citation book cs1"><a href="/wiki/Anthony_C._Davison" title="Anthony C. Davison">Davison, A. C.</a>; <a href="/wiki/David_V._Hinkley" class="mw-redirect" title="David V. Hinkley">Hinkley, D. V.</a> (1997). <i>Bootstrap methods and their application</i>. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57391-2" title="Special:BookSources/0-521-57391-2"><bdi>0-521-57391-2</bdi></a>. <a rel="nofollow" class="external text" href="http://statwww.epfl.ch/davison/BMA/library.html">software</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bootstrap+methods+and+their+application&rft.series=Cambridge+Series+in+Statistical+and+Probabilistic+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-57391-2&rft.aulast=Davison&rft.aufirst=A.+C.&rft.au=Hinkley%2C+D.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Hall1988-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hall1988_42-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall1988" class="citation journal cs1">Hall P (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350933">"Theoretical comparison of bootstrap confidence intervals"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>16</b> (3): 927–953. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350933">10.1214/aos/1176350933</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2241604">2241604</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Theoretical+comparison+of+bootstrap+confidence+intervals&rft.volume=16&rft.issue=3&rft.pages=927-953&rft.date=1988&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176350933&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2241604%23id-name%3DJSTOR&rft.aulast=Hall&rft.aufirst=Peter&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176350933&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-hesterberg2014teachers-43"><span class="mw-cite-backlink">^ <a href="#cite_ref-hesterberg2014teachers_43-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hesterberg2014teachers_43-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hesterberg2014teachers_43-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHesterberg,_Tim_C2014" class="citation arxiv cs1">Hesterberg, Tim C (2014). "What Teachers Should Know about the Bootstrap: Resampling in the Undergraduate Statistics Curriculum". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.5279">1411.5279</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/stat.OT">stat.OT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=What+Teachers+Should+Know+about+the+Bootstrap%3A+Resampling+in+the+Undergraduate+Statistics+Curriculum&rft.date=2014&rft_id=info%3Aarxiv%2F1411.5279&rft.au=Hesterberg%2C+Tim+C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron,_B.1982" class="citation book cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron, B.</a> (1982). <i>The jackknife, the bootstrap, and other resampling plans</i>. Vol. 38. Society of Industrial and Applied Mathematics CBMS-NSF Monographs. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-179-7" title="Special:BookSources/0-89871-179-7"><bdi>0-89871-179-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+jackknife%2C+the+bootstrap%2C+and+other+resampling+plans&rft.pub=Society+of+Industrial+and+Applied+Mathematics+CBMS-NSF+Monographs&rft.date=1982&rft.isbn=0-89871-179-7&rft.au=Efron%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-DAEE-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-DAEE_45-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScheiner1998" class="citation book cs1">Scheiner, S. (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/designanalysisof0000unse"><i>Design and Analysis of Ecological Experiments</i></a></span>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0412035618" title="Special:BookSources/0412035618"><bdi>0412035618</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+and+Analysis+of+Ecological+Experiments&rft.pub=CRC+Press&rft.date=1998&rft.isbn=0412035618&rft.aulast=Scheiner&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdesignanalysisof0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span> Ch13, p300</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRice" class="citation book cs1">Rice, John. <i>Mathematical Statistics and Data Analysis</i> (2 ed.). p. 272.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Statistics+and+Data+Analysis&rft.pages=272&rft.edition=2&rft.aulast=Rice&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span> "Although this direct equation of quantiles of the bootstrap sampling distribution with confidence limits may seem initially appealing, it’s rationale is somewhat obscure."</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.stat.columbia.edu/~gelman/book/data/">Data from examples in <i>Bayesian Data Analysis</i></a></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChiharaHesterberg2018" class="citation book cs1">Chihara, Laura; Hesterberg, Tim (3 August 2018). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9781119505969"><i>Mathematical Statistics with Resampling and R</i></a> (2nd ed.). John Wiley & Sons, Inc. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781119505969">10.1002/9781119505969</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781119416548" title="Special:BookSources/9781119416548"><bdi>9781119416548</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:60138121">60138121</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Statistics+with+Resampling+and+R&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2018-08-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A60138121%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2F9781119505969&rft.isbn=9781119416548&rft.aulast=Chihara&rft.aufirst=Laura&rft.au=Hesterberg%2C+Tim&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fbook%2F10.1002%2F9781119505969&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Voinov, Vassily [G.]; Nikulin, Mikhail [S.] (1993). Unbiased estimators and their applications. Vol. 1: Univariate case. Dordrect: Kluwer Academic Publishers. ISBN 0-7923-2382-3.</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung1990" class="citation journal cs1">Young, G. A. (July 1990). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1111/j.2517-6161.1990.tb01801.x">"Alternative Smoothed Bootstraps"</a>. <i>Journal of the Royal Statistical Society, Series B (Methodological)</i>. <b>52</b> (3): 477–484. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.2517-6161.1990.tb01801.x">10.1111/j.2517-6161.1990.tb01801.x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0035-9246">0035-9246</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B+%28Methodological%29&rft.atitle=Alternative+Smoothed+Bootstraps&rft.volume=52&rft.issue=3&rft.pages=477-484&rft.date=1990-07&rft_id=info%3Adoi%2F10.1111%2Fj.2517-6161.1990.tb01801.x&rft.issn=0035-9246&rft.aulast=Young&rft.aufirst=G.+A.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1111%2Fj.2517-6161.1990.tb01801.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_VaartWellner1996" class="citation book cs1"><a href="/wiki/Aad_van_der_Vaart" title="Aad van der Vaart">van der Vaart AW</a>, <a href="/wiki/Jon_A._Wellner" title="Jon A. Wellner">Wellner JA</a> (1996). <i>Weak Convergence and Empirical Processes With Applications to Statistics</i>. New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Science+Business Media</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-2547-6" title="Special:BookSources/978-1-4757-2547-6"><bdi>978-1-4757-2547-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Weak+Convergence+and+Empirical+Processes+With+Applications+to+Statistics&rft.place=New+York&rft.pub=Springer+Science%2BBusiness+Media&rft.date=1996&rft.isbn=978-1-4757-2547-6&rft.aulast=van+der+Vaart&rft.aufirst=Aad+W.&rft.au=Wellner%2C+Jon+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-kosorok2008-52"><span class="mw-cite-backlink">^ <a href="#cite_ref-kosorok2008_52-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kosorok2008_52-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKosorok2008" class="citation book cs1">Kosorok MR (2008). <i>Introduction to Empirical Processes and Semiparametric Inference</i>. New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Science+Business Media</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-74977-8" title="Special:BookSources/978-0-387-74977-8"><bdi>978-0-387-74977-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Empirical+Processes+and+Semiparametric+Inference&rft.place=New+York&rft.pub=Springer+Science%2BBusiness+Media&rft.date=2008&rft.isbn=978-0-387-74977-8&rft.aulast=Kosorok&rft.aufirst=Michael+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Vaart1998" class="citation book cs1"><a href="/wiki/Aad_van_der_Vaart" title="Aad van der Vaart">van der Vaart AW</a> (1998). <i>Asymptotic Statistics</i>. Cambridge Series in Statistical and Probabilistic Mathematics. New York: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 329. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78450-4" title="Special:BookSources/978-0-521-78450-4"><bdi>978-0-521-78450-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asymptotic+Statistics&rft.place=New+York&rft.series=Cambridge+Series+in+Statistical+and+Probabilistic+Mathematics&rft.pages=329&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-78450-4&rft.aulast=van+der+Vaart&rft.aufirst=Aad+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMammen1992" class="citation book cs1">Mammen E (1992). <i>When Does Bootstrap Work?: Aysmptotic Results and Simulations</i>. Lecture Notes in Statistics. Vol. 57. New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97867-3" title="Special:BookSources/978-0-387-97867-3"><bdi>978-0-387-97867-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=When+Does+Bootstrap+Work%3F%3A+Aysmptotic+Results+and+Simulations&rft.place=New+York&rft.series=Lecture+Notes+in+Statistics&rft.pub=Springer-Verlag&rft.date=1992&rft.isbn=978-0-387-97867-3&rft.aulast=Mammen&rft.aufirst=Enno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGregory2023" class="citation web cs1">Gregory, Karl (29 Dec 2023). <a rel="nofollow" class="external text" href="https://people.stat.sc.edu/gregorkb/STAT_824_sp_2023/STAT_824_Lec_11_supplement.pdf">"Some results based on the Lindeberg central limit theorem"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">29 Dec</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Some+results+based+on+the+Lindeberg+central+limit+theorem&rft.date=2023-12-29&rft.aulast=Gregory&rft.aufirst=Karl&rft_id=https%3A%2F%2Fpeople.stat.sc.edu%2Fgregorkb%2FSTAT_824_sp_2023%2FSTAT_824_Lec_11_supplement.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrapping_(statistics)&action=edit&section=44" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiaconisEfron1983" class="citation journal cs1"><a href="/wiki/Persi_Diaconis" title="Persi Diaconis">Diaconis P</a>, <a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (May 1983). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160313214811/https://statistics.stanford.edu/sites/default/files/BIO%2083.pdf">"Computer-intensive methods in statistics"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>248</b> (5): 116–130. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983SciAm.248e.116D">1983SciAm.248e.116D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0583-116">10.1038/scientificamerican0583-116</a>. Archived from <a rel="nofollow" class="external text" href="https://statistics.stanford.edu/sites/default/files/BIO%2083.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-01-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Computer-intensive+methods+in+statistics&rft.volume=248&rft.issue=5&rft.pages=116-130&rft.date=1983-05&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0583-116&rft_id=info%3Abibcode%2F1983SciAm.248e.116D&rft.aulast=Diaconis&rft.aufirst=Persi&rft.au=Efron%2C+Bradley&rft_id=https%3A%2F%2Fstatistics.stanford.edu%2Fsites%2Fdefault%2Ffiles%2FBIO%252083.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span> popular-science</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron1981" class="citation journal cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (1981). "Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods". <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>. <b>68</b> (3): 589–599. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F68.3.589">10.1093/biomet/68.3.589</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2335441">2335441</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Nonparametric+estimates+of+standard+error%3A+The+jackknife%2C+the+bootstrap+and+other+methods&rft.volume=68&rft.issue=3&rft.pages=589-599&rft.date=1981&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F68.3.589&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2335441%23id-name%3DJSTOR&rft.aulast=Efron&rft.aufirst=Bradley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHesterbergMooreMonaghanClipson2005" class="citation book cs1">Hesterberg T, Moore DS, Monaghan S, Clipson A, Epstein R (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060215221403/http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf">"Bootstrap methods and permutation tests"</a> <span class="cs1-format">(PDF)</span>. In <a href="/wiki/David_S._Moore" title="David S. Moore">David S. Moore</a>, George McCabe (eds.). <i>Introduction to the Practice of Statistics</i>. <a rel="nofollow" class="external text" href="http://www.insightful.com/Hesterberg/bootstrap">software</a>. Archived from <a rel="nofollow" class="external text" href="http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-02-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-03-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bootstrap+methods+and+permutation+tests&rft.btitle=Introduction+to+the+Practice+of+Statistics&rft.date=2005&rft.aulast=Hesterberg&rft.aufirst=Tim&rft.au=Moore%2C+David+S.&rft.au=Monaghan%2C+Shaun&rft.au=Clipson%2C+Ashley&rft.au=Epstein%2C+Rachel&rft_id=http%3A%2F%2Fbcs.whfreeman.com%2Fips5e%2Fcontent%2Fcat_080%2Fpdf%2Fmoore14.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron1979" class="citation journal cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (1979). <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aos/1176344552">"Bootstrap methods: Another look at the jackknife"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>7</b>: 1–26. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">10.1214/aos/1176344552</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Bootstrap+methods%3A+Another+look+at+the+jackknife&rft.volume=7&rft.pages=1-26&rft.date=1979&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176344552&rft.aulast=Efron&rft.aufirst=Bradley&rft_id=http%3A%2F%2Fprojecteuclid.org%2FDPubS%2FRepository%2F1.0%2FDisseminate%3Fview%3Dbody%26id%3Dpdf_1%26handle%3Deuclid.aos%2F1176344552&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron1982" class="citation book cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a> (1982). <i>The Jackknife, the Bootstrap, and Other Resampling Plans</i>. Society of Industrial and Applied Mathematics CBMS-NSF Monographs. Vol. 38. Philadelphia, US: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Jackknife%2C+the+Bootstrap%2C+and+Other+Resampling+Plans&rft.place=Philadelphia%2C+US&rft.series=Society+of+Industrial+and+Applied+Mathematics+CBMS-NSF+Monographs&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1982&rft.aulast=Efron&rft.aufirst=Bradley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfronTibshirani1993" class="citation book cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron B</a>, <a href="/wiki/Robert_Tibshirani" title="Robert Tibshirani">Tibshirani RJ</a> (1993). <i>An Introduction to the Bootstrap</i>. Monographs on Statistics and Applied Probability. Vol. 57. Boca Raton, US: <a href="/wiki/Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>. <a rel="nofollow" class="external text" href="https://archive.today/20120712124533/http://lib.stat.cmu.edu/S/bootstrap.funs">software</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Bootstrap&rft.place=Boca+Raton%2C+US&rft.series=Monographs+on+Statistics+and+Applied+Probability&rft.pub=Chapman+%26+Hall&rft.date=1993&rft.aulast=Efron&rft.aufirst=Bradley&rft.au=Tibshirani%2C+Robert+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisonHinley1997" class="citation book cs1">Davison AC, Hinley DV (1997). <i>Bootstrap Methods and their Application</i>. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780511802843" title="Special:BookSources/9780511802843"><bdi>9780511802843</bdi></a>. <a rel="nofollow" class="external text" href="http://statwww.epfl.ch/davison/BMA/library.html">software</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bootstrap+Methods+and+their+Application&rft.place=Cambridge&rft.series=Cambridge+Series+in+Statistical+and+Probabilistic+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=9780511802843&rft.aulast=Davison&rft.aufirst=A.+C.&rft.au=Hinley%2C+D.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooneyDuval1993" class="citation book cs1 cs1-prop-long-vol">Mooney CZ, Duval RD (1993). <i>Bootstrapping: A Nonparametric Approach to Statistical Inference</i>. Sage University Paper Series on Quantitative Applications in the Social Sciences. Vol. 07–095. Newbury Park, US: <a href="/wiki/SAGE_Publishing" class="mw-redirect" title="SAGE Publishing">Sage</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bootstrapping%3A+A+Nonparametric+Approach+to+Statistical+Inference&rft.place=Newbury+Park%2C+US&rft.series=Sage+University+Paper+Series+on+Quantitative+Applications+in+the+Social+Sciences&rft.pub=Sage&rft.date=1993&rft.aulast=Mooney&rft.aufirst=Christian+Z.&rft.au=Duval%2C+Robert+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWrightLondonField2011" class="citation journal cs1">Wright D, London K, Field AP (2011). "Using bootstrap estimation and the plug-in principle for clinical psychology data". <i>Journal of Experimental Psychopathology</i>. <b>2</b> (2): 252–270. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5127%2Fjep.013611">10.5127/jep.013611</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Experimental+Psychopathology&rft.atitle=Using+bootstrap+estimation+and+the+plug-in+principle+for+clinical+psychology+data&rft.volume=2&rft.issue=2&rft.pages=252-270&rft.date=2011&rft_id=info%3Adoi%2F10.5127%2Fjep.013611&rft.aulast=Wright&rft.aufirst=Daniel&rft.au=London%2C+Kathy&rft.au=Field%2C+Andy+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGong1986" class="citation journal cs1">Gong G (1986). "Cross-validation, the jackknife, and the bootstrap: Excess error estimation in forward logistic regression". <i>Journal of the American Statistical Association</i>. <b>81</b> (393): 108–113. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1986.10478245">10.1080/01621459.1986.10478245</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Cross-validation%2C+the+jackknife%2C+and+the+bootstrap%3A+Excess+error+estimation+in+forward+logistic+regression&rft.volume=81&rft.issue=393&rft.pages=108-113&rft.date=1986&rft_id=info%3Adoi%2F10.1080%2F01621459.1986.10478245&rft.aulast=Gong&rft.aufirst=Gail&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrapping+%28statistics%29" class="Z3988"></span></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Other names that Efron's colleagues suggested for the "bootstrap" method were: <i>Swiss Army Knife</i>, <i>Meat Axe</i>, <i>Swan-Dive</i>, <i>Jack-Rabbit</i>, and <i>Shotgun</i>.<sup id="cite_ref-Efron1979_4-1" class="reference"><a href="#cite_note-Efron1979-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a class="mw-selflink selflink">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a class="mw-selflink selflink">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a 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