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assumptions</span> </div> </a> <button aria-controls="toc-Models_and_assumptions-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 Models and assumptions subsection</span> </button> <ul id="toc-Models_and_assumptions-sublist" class="vector-toc-list"> <li id="toc-Degree_of_models/assumptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Degree_of_models/assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Degree of models/assumptions</span> </div> </a> <ul id="toc-Degree_of_models/assumptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Importance_of_valid_models/assumptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Importance_of_valid_models/assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Importance of valid models/assumptions</span> </div> </a> <ul id="toc-Importance_of_valid_models/assumptions-sublist" class="vector-toc-list"> <li id="toc-Approximate_distributions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Approximate_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Approximate distributions</span> </div> </a> <ul id="toc-Approximate_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Randomization-based_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Randomization-based_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Randomization-based models</span> </div> </a> <ul id="toc-Randomization-based_models-sublist" class="vector-toc-list"> <li id="toc-Model-based_analysis_of_randomized_experiments" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Model-based_analysis_of_randomized_experiments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Model-based analysis of randomized experiments</span> </div> </a> <ul id="toc-Model-based_analysis_of_randomized_experiments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model-free_randomization_inference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Model-free_randomization_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Model-free randomization inference</span> </div> </a> <ul id="toc-Model-free_randomization_inference-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Paradigms_for_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Paradigms_for_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Paradigms for inference</span> </div> </a> <button aria-controls="toc-Paradigms_for_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 Paradigms for inference subsection</span> </button> <ul id="toc-Paradigms_for_inference-sublist" class="vector-toc-list"> <li id="toc-Frequentist_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frequentist_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Frequentist inference</span> </div> </a> <ul id="toc-Frequentist_inference-sublist" class="vector-toc-list"> <li id="toc-Examples_of_frequentist_inference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_of_frequentist_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Examples of frequentist inference</span> </div> </a> <ul id="toc-Examples_of_frequentist_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frequentist_inference,_objectivity,_and_decision_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Frequentist_inference,_objectivity,_and_decision_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Frequentist inference, objectivity, and decision theory</span> </div> </a> <ul id="toc-Frequentist_inference,_objectivity,_and_decision_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bayesian_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bayesian_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Bayesian inference</span> </div> </a> <ul id="toc-Bayesian_inference-sublist" class="vector-toc-list"> <li id="toc-Examples_of_Bayesian_inference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_of_Bayesian_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Examples of Bayesian inference</span> </div> </a> <ul id="toc-Examples_of_Bayesian_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bayesian_inference,_subjectivity_and_decision_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bayesian_inference,_subjectivity_and_decision_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Bayesian inference, subjectivity and decision theory</span> </div> </a> <ul id="toc-Bayesian_inference,_subjectivity_and_decision_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Likelihood-based_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Likelihood-based_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Likelihood-based inference</span> </div> </a> <ul id="toc-Likelihood-based_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-AIC-based_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#AIC-based_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>AIC-based inference</span> </div> </a> <ul id="toc-AIC-based_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_paradigms_for_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_paradigms_for_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Other paradigms for inference</span> </div> </a> <ul id="toc-Other_paradigms_for_inference-sublist" class="vector-toc-list"> <li id="toc-Minimum_description_length" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Minimum_description_length"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.1</span> <span>Minimum description length</span> </div> </a> <ul id="toc-Minimum_description_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fiducial_inference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fiducial_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.2</span> <span>Fiducial inference</span> </div> </a> <ul id="toc-Fiducial_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structural_inference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Structural_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.3</span> <span>Structural inference</span> </div> </a> <ul id="toc-Structural_inference-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Inference_topics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inference_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inference topics</span> </div> </a> <ul id="toc-Inference_topics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Predictive_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Predictive_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Predictive inference</span> </div> </a> <ul id="toc-Predictive_inference-sublist" class="vector-toc-list"> </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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </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">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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">Statistical inference</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 31 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-31" 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">31 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D8%AF%D9%84%D8%A7%D9%84_%D8%A5%D8%AD%D8%B5%D8%A7%D8%A6%D9%8A" title="استدلال إحصائي – Arabic" lang="ar" hreflang="ar" data-title="استدلال إحصائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Infer%C3%A8ncia_estad%C3%ADstica" title="Inferència estadística – Catalan" lang="ca" hreflang="ca" data-title="Inferència estadística" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ystadegaeth_gasgliadol" title="Ystadegaeth gasgliadol – Welsh" lang="cy" hreflang="cy" data-title="Ystadegaeth gasgliadol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Statistische_Inferenz" title="Statistische Inferenz – German" lang="de" hreflang="de" data-title="Statistische Inferenz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estad%C3%ADstica_inferencial" title="Estadística inferencial – Spanish" lang="es" hreflang="es" data-title="Estadística inferencial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Inferentzia_estatistiko" title="Inferentzia estatistiko – Basque" lang="eu" hreflang="eu" data-title="Inferentzia estatistiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%86%D8%A8%D8%A7%D8%B7_%D8%A2%D9%85%D8%A7%D8%B1%DB%8C" title="استنباط آماری – Persian" lang="fa" hreflang="fa" data-title="استنباط آماری" 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/Inf%C3%A9rence_statistique" title="Inférence statistique – French" lang="fr" hreflang="fr" data-title="Inférence statistique" 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/%ED%86%B5%EA%B3%84%EC%A0%81_%EC%B6%94%EB%A1%A0" title="통계적 추론 – Korean" lang="ko" hreflang="ko" data-title="통계적 추론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="सांख्यिकीय अनुमिति – Hindi" lang="hi" hreflang="hi" data-title="सांख्यिकीय अनुमिति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Statistika_inferensial" title="Statistika inferensial – Indonesian" lang="id" hreflang="id" data-title="Statistika inferensial" 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/Inferenza_statistica" title="Inferenza statistica – Italian" lang="it" hreflang="it" data-title="Inferenza 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/%D7%94%D7%A1%D7%A7%D7%94_%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%AA" title="הסקה סטטיסטית – Hebrew" lang="he" hreflang="he" data-title="הסקה סטטיסטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Statistega_inferenziala" title="Statistega inferenziala – Lombard" lang="lmo" hreflang="lmo" data-title="Statistega inferenziala" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inferentie_(statistiek)" title="Inferentie (statistiek) – Dutch" lang="nl" hreflang="nl" data-title="Inferentie (statistiek)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8E%A8%E8%A8%88%E7%B5%B1%E8%A8%88%E5%AD%A6" title="推計統計学 – Japanese" lang="ja" hreflang="ja" data-title="推計統計学" 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/Statistisk_inferens" title="Statistisk inferens – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Statistisk inferens" 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/Wnioskowanie_statystyczne" title="Wnioskowanie statystyczne – Polish" lang="pl" hreflang="pl" data-title="Wnioskowanie statystyczne" 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/Infer%C3%AAncia_estat%C3%ADstica" title="Inferência estatística – Portuguese" lang="pt" hreflang="pt" data-title="Inferência 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%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B2%D1%8B%D0%B2%D0%BE%D0%B4" title="Статистический вывод – Russian" lang="ru" hreflang="ru" data-title="Статистический вывод" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="सांख्यिकीय अनुमान – Sanskrit" lang="sa" hreflang="sa" data-title="सांख्यिकीय अनुमान" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Inference_(statistics)" title="Inference (statistics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Inference (statistics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Research" title="Category:Research">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Research" title="Research">Research</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size mw-image-border" typeof="mw:File/Frameless"><a href="/wiki/File:Humanit%C3%A9s_Num%C3%A9riques.JPG" class="mw-file-description"><img alt="A laptop computer next to archival materials" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Humanit%C3%A9s_Num%C3%A9riques.JPG/220px-Humanit%C3%A9s_Num%C3%A9riques.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Humanit%C3%A9s_Num%C3%A9riques.JPG/330px-Humanit%C3%A9s_Num%C3%A9riques.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Humanit%C3%A9s_Num%C3%A9riques.JPG/440px-Humanit%C3%A9s_Num%C3%A9riques.JPG 2x" data-file-width="3264" data-file-height="2448" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)"><a href="/wiki/Research_design" title="Research design">Research design</a></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Research_ethics" title="Research ethics">Ethics</a></li> <li><a href="/wiki/Research_proposal" title="Research proposal">Proposal</a></li> <li><a href="/wiki/Research_question" title="Research question">Question</a></li> <li><a href="/wiki/Academic_writing" title="Academic writing">Writing</a> <ul><li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Citation" title="Citation">Referencing</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)">Research strategy</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Interdisciplinarity" title="Interdisciplinarity">Interdisciplinary</a></li> <li><a href="/wiki/Multimethodology" title="Multimethodology">Multimethodology</a></li> <li><a href="/wiki/Qualitative_research" title="Qualitative research">Qualitative</a> <ul><li><a href="/wiki/Art-based_research" title="Art-based research">Art-based</a></li></ul></li> <li><a href="/wiki/Quantitative_research" title="Quantitative research">Quantitative</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_schools_of_philosophy" class="mw-redirect" title="List of schools of philosophy">Philosophical schools</a></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Antipositivism" title="Antipositivism">Antipositivism</a></li> <li><a href="/wiki/Constructivism_(philosophy_of_science)" title="Constructivism (philosophy of science)"> Constructivism</a></li> <li><a href="/wiki/Critical_rationalism" title="Critical rationalism">Critical rationalism</a></li> <li><a href="/wiki/Empiricism" title="Empiricism">Empiricism</a></li> <li><a href="/wiki/Fallibilism" title="Fallibilism">Fallibilism</a></li> <li><a href="/wiki/Positivism" title="Positivism">Positivism</a></li> <li><a href="/wiki/Postpositivism" title="Postpositivism">Postpositivism</a></li> <li><a href="/wiki/Pragmatism" title="Pragmatism">Pragmatism</a></li> <li><a href="/wiki/Philosophical_realism" title="Philosophical realism">Realism</a> <ul><li><a href="/wiki/Critical_realism_(philosophy_of_the_social_sciences)" title="Critical realism (philosophy of the social sciences)">Critical realism</a></li> <li><a href="/wiki/Subtle_realism" title="Subtle realism">Subtle realism</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)"><a href="/wiki/Methodology" title="Methodology">Methodology</a></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Action_research" title="Action research">Action research</a></li> <li><a href="/wiki/Art_methodology" title="Art methodology">Art methodology</a></li> <li><a href="/wiki/Critical_theory" title="Critical theory">Critical theory</a></li> <li><a href="/wiki/Grounded_theory" title="Grounded theory">Grounded theory</a></li> <li><a href="/wiki/Hermeneutics" title="Hermeneutics">Hermeneutics</a></li> <li><a href="/wiki/Historiography" title="Historiography">Historiography</a></li> <li><a href="/wiki/Human_subject_research" title="Human subject research">Human subject research</a></li> <li><a href="/wiki/Narrative_inquiry" title="Narrative inquiry">Narrative inquiry</a></li> <li><a href="/wiki/Phenomenology_(sociology)" title="Phenomenology (sociology)">Phenomenology</a></li> <li><a href="/wiki/Pragmatism" title="Pragmatism">Pragmatism</a></li> <li><a href="/wiki/Scientific_method" title="Scientific method">Scientific method</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)">Methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Analysis" title="Analysis">Analysis</a></li> <li><a href="/wiki/Case_study" title="Case study">Case study</a></li> <li><a href="/wiki/Content_analysis" title="Content analysis">Content analysis</a></li> <li><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></li> <li><a href="/wiki/Discourse_analysis" title="Discourse analysis">Discourse analysis</a></li> <li><a href="/wiki/Ethnography" title="Ethnography">Ethnography</a> <ul><li><a href="/wiki/Autoethnography" title="Autoethnography">Autoethnography</a></li></ul></li> <li><a href="/wiki/Experiment" title="Experiment">Experiment</a> <ul><li><a href="/wiki/Field_experiment" title="Field experiment">Field experiment</a></li> <li><a href="/wiki/Social_experiment" title="Social experiment">Social experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul></li> <li><a href="/wiki/Field_research" title="Field research">Field research</a></li> <li><a href="/wiki/Historical_method" title="Historical method">Historical method</a></li> <li><a class="mw-selflink selflink">Inferential statistics</a></li> <li><a href="/wiki/Interview" title="Interview">Interviews</a></li> <li><a href="/wiki/Cartography" title="Cartography">Mapping</a> <ul><li><a href="/wiki/Cultural_mapping" title="Cultural mapping">Cultural mapping</a></li></ul></li> <li><a href="/wiki/Phenomenography" title="Phenomenography">Phenomenography</a></li> <li><a href="/wiki/Secondary_research" title="Secondary research">Secondary research</a> <ul><li><a href="/wiki/Bibliometrics" title="Bibliometrics">Bibliometrics</a></li> <li><a href="/wiki/Literature_review" title="Literature review">Literature review</a></li> <li><a href="/wiki/Meta-analysis" title="Meta-analysis">Meta-analysis</a></li> <li><a href="/wiki/Scoping_review" class="mw-redirect" title="Scoping review">Scoping review</a></li> <li><a href="/wiki/Systematic_review" title="Systematic review">Systematic review</a></li></ul></li> <li><a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a> <ul><li><a href="/wiki/Simulation" title="Simulation">Simulation</a></li></ul></li> <li><a href="/wiki/Survey_(human_research)" title="Survey (human research)">Survey</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="padding-bottom:0;text-align:center;;color: var(--color-base)">Tools and software</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Category:Argument_technology" title="Category:Argument technology">Argument technology</a></li> <li><a href="/wiki/Category:GIS_software" title="Category:GIS software">Geographic information system software</a></li> <li><a href="/wiki/Category:Library_and_information_science_software" title="Category:Library and information science software">Library and information science software</a> <ul><li><a href="/wiki/Category:Bibliometrics_software" title="Category:Bibliometrics software">Bibliometrics</a></li> <li><a href="/wiki/Category:Reference_management_software" title="Category:Reference management software">Reference management</a></li></ul></li> <li><a href="/wiki/Category:Science_software" title="Category:Science software">Science software</a> <ul><li><a href="/wiki/Category:QDA_software" title="Category:QDA software">Qualitative data analysis</a></li> <li><a href="/wiki/Category:Scientific_simulation_software" title="Category:Scientific simulation software">Simulation</a></li> <li><a href="/wiki/Category:Statistical_software" title="Category:Statistical software">Statistics</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="padding-top: 0.1em; font-weight: bold;"> <a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Research_sidebar" title="Template:Research sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Research_sidebar" title="Template talk:Research sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Research_sidebar" title="Special:EditPage/Template:Research sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Process of using data analysis</div> <p><b>Statistical inference</b> is the process of using <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a> to infer properties of an underlying <a href="/wiki/Probability_distribution" title="Probability distribution">distribution of probability</a>.<sup id="cite_ref-Oxford_1-0" class="reference"><a href="#cite_note-Oxford-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <b>Inferential statistical analysis</b> infers properties of a <a href="/wiki/Statistical_population" title="Statistical population">population</a>, for example by <b>testing hypotheses</b> and deriving estimates. It is assumed that the observed data set is <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampled</a> from a larger population. </p><p><b>Inferential statistics</b> can be contrasted with <a href="/wiki/Descriptive_statistics" title="Descriptive statistics">descriptive statistics</a>. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, the term <i>inference</i> is sometimes used instead to mean "make a prediction, by evaluating an already trained model";<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> in this context inferring properties of the model is referred to as <i>training</i> or <i>learning</i> (rather than <i>inference</i>), and using a model for prediction is referred to as <i>inference</i> (instead of <i>prediction</i>); see also <a href="/wiki/Predictive_inference" class="mw-redirect" title="Predictive inference">predictive inference</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Statistical inference makes propositions about a population, using data drawn from the population with some form of <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampling</a>. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) <a href="/wiki/Model_selection" title="Model selection">selecting</a> a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> of the process that generates the data and (second) deducing propositions from the model.<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> </p><p>Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Relatedly, <a href="/wiki/David_Cox_(statistician)" title="David Cox (statistician)">Sir David Cox</a> has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".<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> </p><p>The <a href="/wiki/Logical_consequence" title="Logical consequence">conclusion</a> of a statistical inference is a statistical <a href="/wiki/Proposition" title="Proposition">proposition</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Some common forms of statistical proposition are the following: </p> <ul><li>a <a href="/wiki/Point_estimate" class="mw-redirect" title="Point estimate">point estimate</a>, i.e. a particular value that best approximates some parameter of interest;</li> <li>an <a href="/wiki/Interval_estimate" class="mw-redirect" title="Interval estimate">interval estimate</a>, e.g. a <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> (or set estimate), i.e. an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the <a href="/wiki/Frequency_probability" class="mw-redirect" title="Frequency probability">probability</a> at the stated <a href="/wiki/Confidence_level" class="mw-redirect" title="Confidence level">confidence level</a>;</li> <li>a <a href="/wiki/Credible_intervals" class="mw-redirect" title="Credible intervals">credible interval</a>, i.e. a set of values containing, for example, 95% of posterior belief;</li> <li>rejection of a <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">hypothesis</a>;<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">clustering</a> or <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a> of data points into groups.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Models_and_assumptions">Models and assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=2" title="Edit section: Models and assumptions"><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 articles: <a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> and <a href="/wiki/Statistical_assumptions" class="mw-redirect" title="Statistical assumptions">Statistical assumptions</a></div> <p>Any statistical inference requires some assumptions. A <b>statistical model</b> is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference.<sup id="cite_ref-Cox20062_8-0" class="reference"><a href="#cite_note-Cox20062-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Descriptive_statistic" class="mw-redirect" title="Descriptive statistic">Descriptive statistics</a> are typically used as a preliminary step before more formal inferences are drawn.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Degree_of_models/assumptions"><span id="Degree_of_models.2Fassumptions"></span>Degree of models/assumptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=3" title="Edit section: Degree of models/assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Statisticians distinguish between three levels of modeling assumptions; </p> <ul><li><b><a href="/wiki/Parametric_model" title="Parametric model">Fully parametric</a></b>: The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters.<sup id="cite_ref-Cox20062_8-1" class="reference"><a href="#cite_note-Cox20062-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by <a href="/wiki/Simple_random_sample" title="Simple random sample">'simple' random sampling</a>. The family of <a href="/wiki/Generalized_linear_model#Model_components" title="Generalized linear model">generalized linear models</a> is a widely used and flexible class of parametric models.</li> <li><b><a href="/wiki/Nonparametric_statistics#Non-parametric_models" title="Nonparametric statistics"> Non-parametric</a></b>: The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For example, every continuous probability distribution has a median, which may be estimated using the sample median or the <a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann–Sen estimator</a>, which has good properties when the data arise from simple random sampling.</li> <li><b><a href="/wiki/Semiparametric_model" title="Semiparametric model">Semi-parametric</a></b>: This term typically implies assumptions 'in between' fully and non-parametric approaches. For example, one may assume that a population distribution has a finite mean. Furthermore, one may assume that the mean response level in the population depends in a truly linear manner on some covariate (a parametric assumption) but not make any parametric assumption describing the variance around that mean (i.e. about the presence or possible form of any <a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">heteroscedasticity</a>). More generally, semi-parametric models can often be separated into 'structural' and 'random variation' components. One component is treated parametrically and the other non-parametrically. The well-known <a href="/wiki/Cox_model" class="mw-redirect" title="Cox model">Cox model</a> is a set of semi-parametric assumptions.<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. (November 2023)">citation needed</span></a></i>]</sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Importance_of_valid_models/assumptions"><span id="Importance_of_valid_models.2Fassumptions"></span>Importance of valid models/assumptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=4" title="Edit section: Importance of valid models/assumptions"><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">See also: <a href="/wiki/Statistical_model_validation" title="Statistical model validation">Statistical model validation</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Normality_Histogram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Normality_Histogram.png/220px-Normality_Histogram.png" decoding="async" width="220" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Normality_Histogram.png/330px-Normality_Histogram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Normality_Histogram.png/440px-Normality_Histogram.png 2x" data-file-width="1057" data-file-height="621" /></a><figcaption>The above image shows a histogram assessing the assumption of normality, which can be illustrated through the even spread underneath the bell curve.</figcaption></figure> <p>Whatever level of assumption is made, correctly calibrated inference, in general, requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified. </p><p>Incorrect assumptions of <a href="/wiki/Simple_random_sample" title="Simple random sample">'simple' random sampling</a> can invalidate statistical inference.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The use of <b>any</b> parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal."<sup id="cite_ref-Brewer2_14-0" class="reference"><a href="#cite_note-Brewer2-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population."<sup id="cite_ref-Brewer2_14-1" class="reference"><a href="#cite_note-Brewer2-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy-tailed. </p> <div class="mw-heading mw-heading4"><h4 id="Approximate_distributions">Approximate distributions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=5" title="Edit section: Approximate distributions"><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 articles: <a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a>, <a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotic theory (statistics)</a>, and <a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></div> <p>Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these. </p><p>With finite samples, <a href="/wiki/Approximation_theory" title="Approximation theory">approximation results</a> measure how close a limiting distribution approaches the statistic's <a href="/wiki/Sample_distribution" class="mw-redirect" title="Sample distribution">sample distribution</a>: For example, with 10,000 independent samples the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> approximates (to two digits of accuracy) the distribution of the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> for many population distributions, by the <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen theorem</a>.<sup id="cite_ref-JHJ2_15-0" class="reference"><a href="#cite_note-JHJ2-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Yet for many practical purposes, the normal approximation provides a good approximation to the sample-mean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience.<sup id="cite_ref-JHJ2_15-1" class="reference"><a href="#cite_note-JHJ2-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Following Kolmogorov's work in the 1950s, advanced statistics uses <a href="/wiki/Approximation_theory" title="Approximation theory">approximation theory</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> to quantify the error of approximation. In this approach, the <a href="/wiki/Metric_geometry" class="mw-redirect" title="Metric geometry">metric geometry</a> of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> is studied; this approach quantifies approximation error with, for example, the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a>, <a href="/wiki/Bregman_divergence" title="Bregman divergence">Bregman divergence</a>, and the <a href="/wiki/Hellinger_distance" title="Hellinger distance">Hellinger distance</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><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> </p><p>With indefinitely large samples, <a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">limiting results</a> like the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> describe the sample statistic's limiting distribution if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.<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><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><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the <a href="/wiki/Generalized_method_of_moments" title="Generalized method of moments">generalized method of moments</a> and the use of <a href="/wiki/Generalized_estimating_equation" title="Generalized estimating equation">generalized estimating equations</a>, which are popular in <a href="/wiki/Econometrics" title="Econometrics">econometrics</a> and <a href="/wiki/Biostatistics" title="Biostatistics">biostatistics</a>. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation.<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> The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional <a href="/wiki/Statistical_model" title="Statistical model">models</a> with <a href="/wiki/Logarithmically_concave_function" title="Logarithmically concave function">log-concave</a> <a href="/wiki/Likelihood_function" title="Likelihood function">likelihoods</a> (such as with one-parameter <a href="/wiki/Exponential_families" class="mw-redirect" title="Exponential families">exponential families</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Randomization-based_models">Randomization-based models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=6" title="Edit section: Randomization-based models"><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/Randomization" title="Randomization">Randomization</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Random_sample" class="mw-redirect" title="Random sample">Random sample</a> and <a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></div> <p>For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the null-hypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, the randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments.<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><sup id="cite_ref-Hinkelmann_and_Kempthorne2_24-0" class="reference"><a href="#cite_note-Hinkelmann_and_Kempthorne2-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Statistical inference from randomized studies is also more straightforward than many other situations.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><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><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, randomization is also of importance: in <a href="/wiki/Survey_sampling" title="Survey sampling">survey sampling</a>, use of <a href="/wiki/Sampling_without_replacement" class="mw-redirect" title="Sampling without replacement">sampling without replacement</a> ensures the <a href="/wiki/Exchangeability" class="mw-redirect" title="Exchangeability">exchangeability</a> of the sample with the population; in randomized experiments, randomization warrants a <a href="/wiki/Missing_at_random" class="mw-redirect" title="Missing at random">missing at random</a> assumption for <a href="/wiki/Covariate" class="mw-redirect" title="Covariate">covariate</a> information.<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> </p><p>Objective randomization allows properly inductive procedures.<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><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><sup id="cite_ref-FOOTNOTEFreedmanPisaniPurves1978_31-0" class="reference"><a href="#cite_note-FOOTNOTEFreedmanPisaniPurves1978-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><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><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> Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures.<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> (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.<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><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>) Similarly, results from <a href="/wiki/Randomized_experiment" title="Randomized experiment">randomized experiments</a> are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena.<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> However, a good observational study may be better than a bad randomized experiment. </p><p>The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.<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><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><p>However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical. </p> <div class="mw-heading mw-heading4"><h4 id="Model-based_analysis_of_randomized_experiments">Model-based analysis of randomized experiments</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=7" title="Edit section: Model-based analysis of randomized experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is standard practice to refer to a statistical model, e.g., a linear or logistic models, when analyzing data from randomized experiments.<sup id="cite_ref-Dinov_Palanimalai_Khare_Christou_20182_40-0" class="reference"><a href="#cite_note-Dinov_Palanimalai_Khare_Christou_20182-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme.<sup id="cite_ref-Hinkelmann_and_Kempthorne2_24-1" class="reference"><a href="#cite_note-Hinkelmann_and_Kempthorne2-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Model-free_randomization_inference">Model-free randomization inference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=8" title="Edit section: Model-free randomization inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of reality-simplification. The former combine, evolve, ensemble and train algorithms dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations.<sup id="cite_ref-Dinov_Palanimalai_Khare_Christou_2018_42-0" class="reference"><a href="#cite_note-Dinov_Palanimalai_Khare_Christou_2018-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tang_model-based_Model-Free_2019_43-0" class="reference"><a href="#cite_note-Tang_model-based_Model-Free_2019-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>For example, model-free simple linear regression is based either on </p> <ul><li>a <i>random design</i>, where the pairs of observations <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},Y_{1}),(X_{2},Y_{2}),\cdots ,(X_{n},Y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},Y_{1}),(X_{2},Y_{2}),\cdots ,(X_{n},Y_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1108d94da1a53e6a5c5dc5743a59e1a53cbefe16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.22ex; height:2.843ex;" alt="{\displaystyle (X_{1},Y_{1}),(X_{2},Y_{2}),\cdots ,(X_{n},Y_{n})}"></span> are independent and identically distributed (iid), or</li> <li>a <i>deterministic design</i>, where the 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 X_{1},X_{2},\cdots ,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> <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"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\cdots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2cf515b30e40cb1ea724ecbf8dd94428691a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\cdots ,X_{n}}"></span> are deterministic, but the corresponding 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_{1},Y_{2},\cdots ,Y_{n}}"> <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> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{1},Y_{2},\cdots ,Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52889f2a721713f345ddae4187cfa8af5101b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.591ex; height:2.509ex;" alt="{\displaystyle Y_{1},Y_{2},\cdots ,Y_{n}}"></span> are random and independent with a common conditional distribution, 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\left(Y_{j}\leq y|X_{j}=x\right)=D_{x}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≤</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\left(Y_{j}\leq y|X_{j}=x\right)=D_{x}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5739753e0f4ff7becb80315bb535f3834ec0697c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.525ex; height:3.009ex;" alt="{\displaystyle P\left(Y_{j}\leq y|X_{j}=x\right)=D_{x}(y)}"></span>, which is independent of the index <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>.</li></ul> <p>In either case, the model-free randomization inference for features of the common conditional distribution <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_{x}(.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}(.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113b1c267289798bd4a0c62dc51db4ec2d4e2ba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.94ex; height:2.843ex;" alt="{\displaystyle D_{x}(.)}"></span> relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature <i>conditional mean</i>, <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 (x)=E(Y|X=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x)=E(Y|X=x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd6399d74e438553f47cd54fc7392ecbe70e244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.052ex; height:2.843ex;" alt="{\displaystyle \mu (x)=E(Y|X=x)}"></span>, can be consistently estimated via local averaging or local polynomial fitting, under the assumption 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 \mu (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f339251a09ebf15dd50bb751d27b02820f68c545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \mu (x)}"></span> is smooth. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the <i>conditional mean</i>, <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 (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f339251a09ebf15dd50bb751d27b02820f68c545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \mu (x)}"></span>.<sup id="cite_ref-Politis_Model-Free_Inference_2019_44-0" class="reference"><a href="#cite_note-Politis_Model-Free_Inference_2019-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Paradigms_for_inference">Paradigms for inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=9" title="Edit section: Paradigms for inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms. </p><p>Bandyopadhyay & Forster describe four paradigms: The classical (or <a href="/wiki/Frequentist_inference" title="Frequentist inference">frequentist</a>) paradigm, the <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian</a> paradigm, the <a href="/wiki/Likelihoodism" class="mw-redirect" title="Likelihoodism">likelihoodist</a> paradigm, and the <a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaikean-Information Criterion</a>-based paradigm.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Frequentist_inference">Frequentist inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=10" title="Edit section: Frequentist inference"><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/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></div> <p>This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging. </p> <div class="mw-heading mw-heading4"><h4 id="Examples_of_frequentist_inference">Examples of frequentist inference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=11" title="Edit section: Examples of frequentist inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/P-value" title="P-value"><i>p</i>-value</a></li> <li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Null_hypothesis" title="Null hypothesis">Null hypothesis</a> significance testing</li></ul> <div class="mw-heading mw-heading4"><h4 id="Frequentist_inference,_objectivity,_and_decision_theory"><span id="Frequentist_inference.2C_objectivity.2C_and_decision_theory"></span>Frequentist inference, objectivity, and decision theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=12" title="Edit section: Frequentist inference, objectivity, and decision theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One interpretation of <a href="/wiki/Frequentist_inference" title="Frequentist inference">frequentist inference</a> (or classical inference) is that it is applicable only in terms of <a href="/wiki/Frequency_probability" class="mw-redirect" title="Frequency probability">frequency probability</a>; that is, in terms of repeated sampling from a population. However, the approach of Neyman<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> develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e. probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach. </p><p>The frequentist procedures of significance testing and confidence intervals can be constructed without regard to <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility functions</a>. However, some elements of frequentist statistics, such as <a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">statistical decision theory</a>, do incorporate <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility functions</a>.<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 2012)">citation needed</span></a></i>]</sup> In particular, frequentist developments of optimal inference (such as <a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum-variance unbiased estimators</a>, or <a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">uniformly most powerful testing</a>) make use of <a href="/wiki/Loss_function" title="Loss function">loss functions</a>, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property.<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> However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> loss functions, in that they minimize expected loss, and <a href="/wiki/Least_squares" title="Least squares">least squares</a> estimators are optimal under squared error loss functions, in that they minimize expected loss. </p><p>While statisticians using frequentist inference must choose for themselves the parameters of interest, and the <a href="/wiki/Estimators" class="mw-redirect" title="Estimators">estimators</a>/<a href="/wiki/Test_statistic#Common_test_statistics" title="Test statistic">test statistic</a> to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Bayesian_inference">Bayesian inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=13" title="Edit section: Bayesian inference"><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">See also: <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></div> <p>The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate into one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions.<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> There are <a href="/wiki/Bayesian_probability#Justification_of_Bayesian_probabilities" title="Bayesian probability">several different justifications</a> for using the Bayesian approach. </p> <div class="mw-heading mw-heading4"><h4 id="Examples_of_Bayesian_inference">Examples of Bayesian inference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=14" title="Edit section: Examples of Bayesian inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a> for <a href="/wiki/Interval_estimation" title="Interval estimation">interval estimation</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factors</a> for model comparison</li></ul> <div class="mw-heading mw-heading4"><h4 id="Bayesian_inference,_subjectivity_and_decision_theory"><span id="Bayesian_inference.2C_subjectivity_and_decision_theory"></span>Bayesian inference, subjectivity and decision theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=15" title="Edit section: Bayesian inference, subjectivity and decision theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a> need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been <a href="/wiki/Bayesian_probability#Personal_probabilities_and_objective_methods_for_constructing_priors" title="Bayesian probability">proposed</a> but not yet fully developed.) </p><p>Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides <a href="/wiki/Optimal_decision" title="Optimal decision">optimal decisions</a> in a <a href="/wiki/Decision_theory" title="Decision theory">decision theoretic</a> sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) <a href="/wiki/Coherence_(statistics)" title="Coherence (statistics)">incoherent</a>; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be <a href="/wiki/Coherence_(statistics)" title="Coherence (statistics)">coherent</a>. Some advocates of <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> assert that inference <i>must</i> take place in this decision-theoretic framework, and that <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> should not conclude with the evaluation and summarization of posterior beliefs. </p> <div class="mw-heading mw-heading3"><h3 id="Likelihood-based_inference">Likelihood-based inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=16" title="Edit section: Likelihood-based inference"><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/Likelihoodism" class="mw-redirect" title="Likelihoodism">Likelihoodism</a></div><p>Likelihood-based inference is a paradigm used to estimate the parameters of a statistical model based on observed data. <a href="/wiki/Likelihoodism" class="mw-redirect" title="Likelihoodism">Likelihoodism</a> approaches statistics by using the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a>, denoted 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 L(x|\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(x|\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f093bbc3c24bc49321c42d5de9f2cabb6286139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.459ex; height:2.843ex;" alt="{\displaystyle L(x|\theta )}"></span>, quantifies the probability of observing the given data <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>, assuming a specific set of parameter 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </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/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. In likelihood-based inference, the goal is to find the set of parameter values that maximizes the likelihood function, or equivalently, maximizes the probability of observing the given data. </p><p>The process of likelihood-based inference usually involves the following steps: </p> <ol><li>Formulating the statistical model: A statistical model is defined based on the problem at hand, specifying the distributional assumptions and the relationship between the observed data and the unknown parameters. The model can be simple, such as a normal distribution with known variance, or complex, such as a hierarchical model with multiple levels of random effects.</li> <li>Constructing the likelihood function: Given the statistical model, the likelihood function is constructed by evaluating the joint probability density or mass function of the observed data as a function of the unknown parameters. This function represents the probability of observing the data for different values of the parameters.</li> <li>Maximizing the likelihood function: The next step is to find the set of parameter values that maximizes the likelihood function. This can be achieved using optimization techniques such as numerical optimization algorithms. The estimated parameter values, often denoted 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 {\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>, are the <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimates</a> (MLEs).</li> <li>Assessing uncertainty: Once the MLEs are obtained, it is crucial to quantify the uncertainty associated with the parameter estimates. This can be done by calculating <a href="/wiki/Standard_error" title="Standard error">standard errors</a>, confidence intervals, or conducting <a href="/wiki/Hypothesis_test" class="mw-redirect" title="Hypothesis test">hypothesis tests</a> based on asymptotic theory or simulation techniques such as <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrapping</a>.</li> <li>Model checking: After obtaining the parameter estimates and assessing their uncertainty, it is important to assess the adequacy of the statistical model. This involves checking the assumptions made in the model and evaluating the fit of the model to the data using goodness-of-fit tests, residual analysis, or graphical diagnostics.</li> <li>Inference and interpretation: Finally, based on the estimated parameters and model assessment, statistical inference can be performed. This involves drawing conclusions about the population parameters, making predictions, or testing hypotheses based on the estimated model.</li></ol> <div class="mw-heading mw-heading3"><h3 id="AIC-based_inference">AIC-based inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=17" title="Edit section: AIC-based inference"><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/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a></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-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Statistical_inference&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">November 2017</span>)</i></span></div></td></tr></tbody></table> <p>The <i><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a></i> (AIC) is an <a href="/wiki/Estimator" title="Estimator">estimator</a> of the relative quality of <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for <a href="/wiki/Model_selection" title="Model selection">model selection</a>. </p><p>AIC is founded on <a href="/wiki/Information_theory" title="Information theory">information theory</a>: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a> of the model and the simplicity of the model.) </p> <div class="mw-heading mw-heading3"><h3 id="Other_paradigms_for_inference">Other paradigms for inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=18" title="Edit section: Other paradigms for inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Minimum_description_length">Minimum description length</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=19" title="Edit section: Minimum description length"><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/Minimum_description_length" title="Minimum description length">Minimum description length</a></div> <p>The minimum description length (MDL) principle has been developed from ideas in <a href="/wiki/Information_theory" title="Information theory">information theory</a><sup id="cite_ref-Soofi_2000_1349–1353_50-0" class="reference"><a href="#cite_note-Soofi_2000_1349–1353-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> and the theory of <a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>.<sup id="cite_ref-HY_51-0" class="reference"><a href="#cite_note-HY-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> The (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or non-falsifiable "data-generating mechanisms" or <a href="/wiki/Probability_models" class="mw-redirect" title="Probability models">probability models</a> for the data, as might be done in frequentist or Bayesian approaches. </p><p>However, if a "data generating mechanism" does exist in reality, then according to <a href="/wiki/Claude_Shannon" title="Claude Shannon">Shannon</a>'s <a href="/wiki/Source_coding_theorem" class="mw-redirect" title="Source coding theorem">source coding theorem</a> it provides the MDL description of the data, on average and asymptotically.<sup id="cite_ref-HY747_52-0" class="reference"><a href="#cite_note-HY747-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> In minimizing description length (or descriptive complexity), MDL estimation is similar to <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a> and <a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">maximum a posteriori estimation</a> (using <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum-entropy</a> <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian priors</a>). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.<sup id="cite_ref-HY747_52-1" class="reference"><a href="#cite_note-HY747-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JR_53-0" class="reference"><a href="#cite_note-JR-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>The MDL principle has been applied in communication-<a href="/wiki/Coding_theory" title="Coding theory">coding theory</a> in <a href="/wiki/Information_theory" title="Information theory">information theory</a>, in <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a>,<sup id="cite_ref-JR_53-1" class="reference"><a href="#cite_note-JR-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> and in <a href="/wiki/Data_mining" title="Data mining">data mining</a>.<sup id="cite_ref-HY_51-1" class="reference"><a href="#cite_note-HY-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p>The evaluation of MDL-based inferential procedures often uses techniques or criteria from <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Fiducial_inference">Fiducial inference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=20" title="Edit section: Fiducial inference"><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/Fiducial_inference" title="Fiducial inference">Fiducial inference</a></div> <p><a href="/wiki/Fiducial_inference" title="Fiducial inference">Fiducial inference</a> was an approach to statistical inference based on <a href="/wiki/Fiducial_probability" class="mw-redirect" title="Fiducial probability">fiducial probability</a>, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious.<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><sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> However this argument is the same as that which shows<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> that a so-called <a href="/wiki/Confidence_distribution" title="Confidence distribution">confidence distribution</a> is not a valid <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> and, since this has not invalidated the application of <a href="/wiki/Confidence_interval" title="Confidence interval">confidence intervals</a>, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's <a href="/wiki/Fiducial_probability" class="mw-redirect" title="Fiducial probability">fiducial argument</a> as a special case of an inference theory using <a href="/wiki/Upper_and_lower_probabilities" title="Upper and lower probabilities">upper and lower probabilities</a>.<sup id="cite_ref-FOOTNOTEHampel2003_58-0" class="reference"><a href="#cite_note-FOOTNOTEHampel2003-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Structural_inference">Structural inference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=21" title="Edit section: Structural inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Developing ideas of Fisher and of Pitman from 1938 to 1939,<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> <a href="/wiki/George_A._Barnard" class="mw-redirect" title="George A. Barnard">George A. Barnard</a> developed "structural inference" or "pivotal inference",<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> an approach using <a href="/wiki/Haar_measure" title="Haar measure">invariant probabilities</a> on <a href="/wiki/Group_family" title="Group family">group families</a>. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be well-defined and useful. <a href="/wiki/Donald_A._S._Fraser" title="Donald A. S. Fraser">Donald A. S. Fraser</a> developed a general theory for structural inference<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> based on <a href="/wiki/Group_theory" title="Group theory">group theory</a> and applied this to linear models.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Inference_topics">Inference topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=22" title="Edit section: Inference topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The topics below are usually included in the area of <b>statistical inference</b>. </p> <ol><li><a href="/wiki/Statistical_assumptions" class="mw-redirect" title="Statistical assumptions">Statistical assumptions</a></li> <li><a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">Statistical decision theory</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Statistical hypothesis testing</a></li> <li><a href="/w/index.php?title=Revising_opinions_in_statistics&action=edit&redlink=1" class="new" title="Revising opinions in statistics (page does not exist)">Revising opinions in statistics</a></li> <li><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</a>, the <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a>, and <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></li> <li><a href="/wiki/Survey_sampling" title="Survey sampling">Survey sampling</a></li> <li><a href="/wiki/Summarizing_statistical_data" class="mw-redirect" title="Summarizing statistical data">Summarizing statistical data</a></li></ol> <div class="mw-heading mw-heading2"><h2 id="Predictive_inference">Predictive inference <span class="anchor" id="Prediction"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=23" title="Edit section: Predictive inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Predictive inference is an approach to statistical inference that emphasizes the <a href="/wiki/Prediction" title="Prediction">prediction</a> of future observations based on past observations. </p><p>Initially, predictive inference was based on <i>observable</i> parameters and it was the main purpose of studying <a href="/wiki/Probability" title="Probability">probability</a>,<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. (November 2011)">citation needed</span></a></i>]</sup> but it fell out of favor in the 20th century due to a new parametric approach pioneered by <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a>. The approach modeled phenomena as a physical system observed with error (e.g., <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>). De Finetti's idea of <a href="/wiki/Exchangeability" class="mw-redirect" title="Exchangeability">exchangeability</a>—that future observations should behave like past observations—came to the attention of the English-speaking world with the 1974 translation from French of his 1937 paper,<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> and has since been propounded by such statisticians as <a href="/wiki/Seymour_Geisser" title="Seymour Geisser">Seymour Geisser</a>.<sup id="cite_ref-geisser_65-0" class="reference"><a href="#cite_note-geisser-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </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=Statistical_inference&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algorithmic_inference" title="Algorithmic inference">Algorithmic inference</a></li> <li><a href="/wiki/Induction_(philosophy)" class="mw-redirect" title="Induction (philosophy)">Induction (philosophy)</a></li> <li><a href="/wiki/Informal_inferential_reasoning" title="Informal inferential reasoning">Informal inferential reasoning</a></li> <li><a href="/wiki/Information_field_theory" title="Information field theory">Information field theory</a></li> <li><a href="/wiki/Population_proportion" title="Population proportion">Population proportion</a></li> <li><a href="/wiki/Philosophy_of_statistics" title="Philosophy of statistics">Philosophy of statistics</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Predictive_analytics" title="Predictive analytics">Predictive analytics</a></li> <li><a href="/wiki/Predictive_modelling" title="Predictive modelling">Predictive modelling</a></li> <li><a href="/wiki/Stylometry" title="Stylometry">Stylometry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=25" title="Edit section: Notes"><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"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">According to Peirce, acceptance means that inquiry on this question ceases for the time being. In science, all scientific theories are revisable.</span> </li> </ol></div></div> <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=Statistical_inference&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=27" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Oxford-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Oxford_1-0">^</a></b></span> <span class="reference-text">Upton, G., Cook, I. (2008) <i>Oxford Dictionary of Statistics</i>, OUP. <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-954145-4" title="Special:BookSources/978-0-19-954145-4">978-0-19-954145-4</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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://www.tensorflow.org/lite/guide/inference">"TensorFlow Lite inference"</a>. <q>The term <i>inference</i> refers to the process of executing a TensorFlow Lite model on-device in order to make predictions based on input data.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=TensorFlow+Lite+inference&rft_id=https%3A%2F%2Fwww.tensorflow.org%2Flite%2Fguide%2Finference&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></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="CITEREFJohnson2016" class="citation web cs1">Johnson, Richard (12 March 2016). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Statistical_inference">"Statistical Inference"</a>. <i>Encyclopedia of Mathematics</i>. Springer: The European Mathematical Society<span class="reference-accessdate">. Retrieved <span class="nowrap">26 October</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Statistical+Inference&rft.date=2016-03-12&rft.aulast=Johnson&rft.aufirst=Richard&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FStatistical_inference&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Konishi & Kitagawa (2008), p. 75.</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">Cox (2006), p. 197.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</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://www.encyclopediaofmath.org/index.php/Statistical_inference">"Statistical inference - Encyclopedia of Mathematics"</a>. <i>www.encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-01-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.encyclopediaofmath.org&rft.atitle=Statistical+inference+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FStatistical_inference&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-Cox20062-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cox20062_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cox20062_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Cox (2006) page 2</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans2004" class="citation book cs1">Evans, Michael; et al. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hkWK8kFzXWIC"><i>Probability and Statistics: The Science of Uncertainty</i></a>. Freeman and Company. p. 267. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780716747420" title="Special:BookSources/9780716747420"><bdi>9780716747420</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Statistics%3A+The+Science+of+Uncertainty&rft.pages=267&rft.pub=Freeman+and+Company&rft.date=2004&rft.isbn=9780716747420&rft.aulast=Evans&rft.aufirst=Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhkWK8kFzXWIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">van der Vaart, A.W. (1998) <i>Asymptotic Statistics</i> Cambridge University Press. <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/0-521-78450-6" title="Special:BookSources/0-521-78450-6">0-521-78450-6</a> (page 341)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Kruskal 1988</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman, D.A.</a> (2008) "Survival analysis: An Epidemiological hazard?". <i>The American Statistician</i> (2008) 62: 110-119. (Reprinted as Chapter 11 (pages 169–192) of Freedman (2010)).</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Berk, R. (2003) <i>Regression Analysis: A Constructive Critique (Advanced Quantitative Techniques in the Social Sciences) (v. 11)</i> Sage Publications. <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/0-7619-2904-5" title="Special:BookSources/0-7619-2904-5">0-7619-2904-5</a></span> </li> <li id="cite_note-Brewer2-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Brewer2_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Brewer2_14-1"><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="CITEREFBrewer2002" class="citation book cs1">Brewer, Ken (2002). <i>Combined Survey Sampling Inference: Weighing of Basu's Elephants</i>. Hodder Arnold. p. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0340692295" title="Special:BookSources/978-0340692295"><bdi>978-0340692295</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combined+Survey+Sampling+Inference%3A+Weighing+of+Basu%27s+Elephants&rft.pages=6&rft.pub=Hodder+Arnold&rft.date=2002&rft.isbn=978-0340692295&rft.aulast=Brewer&rft.aufirst=Ken&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-JHJ2-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-JHJ2_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-JHJ2_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Jörgen Hoffman-Jörgensen's <i>Probability With a View Towards Statistics</i>, Volume I. Page 399 <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (November 2012)">full citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Le Cam (1986) <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (June 2011)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Erik Torgerson (1991) <i>Comparison of Statistical Experiments</i>, volume 36 of Encyclopedia of Mathematics. Cambridge University Press. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (November 2012)">full citation needed</span></a></i>]</sup></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="CITEREFLiese,_FriedrichMiescke,_Klaus-J.2008" class="citation book cs1">Liese, Friedrich & Miescke, Klaus-J. (2008). <i>Statistical Decision Theory: Estimation, Testing, and Selection</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-73193-3" title="Special:BookSources/978-0-387-73193-3"><bdi>978-0-387-73193-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Decision+Theory%3A+Estimation%2C+Testing%2C+and+Selection&rft.pub=Springer&rft.date=2008&rft.isbn=978-0-387-73193-3&rft.au=Liese%2C+Friedrich&rft.au=Miescke%2C+Klaus-J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" 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">Kolmogorov (1963, p.369): "The frequency concept, based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials".</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">"Indeed, limit theorems '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}"> <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> tends to infinity' are logically devoid of content about what happens at any particular <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>. All they can do is suggest certain approaches whose performance must then be checked on the case at hand." — Le Cam (1986) (page xiv)</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Pfanzagl (1994): "The crucial drawback of asymptotic theory: What we expect from asymptotic theory are results which hold approximately . . . . What asymptotic theory has to offer are limit theorems."(page ix) "What counts for applications are approximations, not limits." (page 188)</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">Pfanzagl (1994) : "By taking a limit theorem as being approximately true for large sample sizes, we commit an error the size of which is unknown. [. . .] Realistic information about the remaining errors may be obtained by simulations." (page ix)</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"><a href="/wiki/Jerzy_Neyman" title="Jerzy Neyman">Neyman, J.</a>(1934) "On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection", <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society" title="Journal of the Royal Statistical Society">Journal of the Royal Statistical Society</a></i>, 97 (4), 557–625 <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/2342192">2342192</a></span> </li> <li id="cite_note-Hinkelmann_and_Kempthorne2-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hinkelmann_and_Kempthorne2_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hinkelmann_and_Kempthorne2_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hinkelmann and Kempthorne(2008) <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (June 2011)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">ASA Guidelines for the first course in statistics for non-statisticians. (available at the ASA website)</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"><a href="/wiki/David_A._Freedman" title="David A. Freedman">David A. Freedman</a> et alia's <i>Statistics</i>.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Moore et al. (2015).</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"><a href="/wiki/Andrew_Gelman" title="Andrew Gelman">Gelman A.</a> et al. (2013). <i>Bayesian Data Analysis</i> (<a href="/wiki/Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>).</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">Peirce (1877-1878)</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">Peirce (1883)</span> </li> <li id="cite_note-FOOTNOTEFreedmanPisaniPurves1978-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFreedmanPisaniPurves1978_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFreedmanPisaniPurves1978">Freedman, Pisani & Purves 1978</a>.</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"><a href="/wiki/David_A._Freedman" title="David A. Freedman">David A. Freedman</a> <i>Statistical Models</i>.</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"><a href="/wiki/C._R._Rao" title="C. R. Rao">Rao, C.R.</a> (1997) <i>Statistics and Truth: Putting Chance to Work</i>, World Scientific. <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/981-02-3111-3" title="Special:BookSources/981-02-3111-3">981-02-3111-3</a></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">Peirce; Freedman; Moore et al. (2015).<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. (March 2010)">citation needed</span></a></i>]</sup></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">Box, G.E.P. and Friends (2006) <i>Improving Almost Anything: Ideas and Essays, Revised Edition</i>, Wiley. <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/978-0-471-72755-2" title="Special:BookSources/978-0-471-72755-2">978-0-471-72755-2</a></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">Cox (2006), p. 196.</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">ASA Guidelines for the first course in statistics for non-statisticians. (available at the ASA website) <ul><li>David A. Freedman et alias <i>Statistics</i>.</li> <li>Moore et al. (2015).</li></ul> </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">Neyman, Jerzy. 1923 [1990]. "On the Application of Probability Theory to AgriculturalExperiments. Essay on Principles. Section 9." <i>Statistical Science</i> 5 (4): 465–472. Trans. <a href="/wiki/Dorota_Dabrowska" title="Dorota Dabrowska">Dorota M. Dabrowska</a> and Terence P. Speed.</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">Hinkelmann & Kempthorne (2008) <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (June 2011)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-Dinov_Palanimalai_Khare_Christou_20182-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dinov_Palanimalai_Khare_Christou_20182_40-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDinovPalanimalaiKhareChristou2018" class="citation journal cs1">Dinov, Ivo; Palanimalai, Selvam; Khare, Ashwini; Christou, Nicolas (2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6155997">"Randomization-based statistical inference: A resampling and simulation infrastructure"</a>. <i>Teaching Statistics</i>. <b>40</b> (2): 64–73. <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%2Ftest.12156">10.1111/test.12156</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/PMC6155997">6155997</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/30270947">30270947</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Teaching+Statistics&rft.atitle=Randomization-based+statistical+inference%3A+A+resampling+and+simulation+infrastructure&rft.volume=40&rft.issue=2&rft.pages=64-73&rft.date=2018&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6155997%23id-name%3DPMC&rft_id=info%3Apmid%2F30270947&rft_id=info%3Adoi%2F10.1111%2Ftest.12156&rft.aulast=Dinov&rft.aufirst=Ivo&rft.au=Palanimalai%2C+Selvam&rft.au=Khare%2C+Ashwini&rft.au=Christou%2C+Nicolas&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6155997&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">Hinkelmann and Kempthorne (2008) Chapter 6.</span> </li> <li id="cite_note-Dinov_Palanimalai_Khare_Christou_2018-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dinov_Palanimalai_Khare_Christou_2018_42-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDinovPalanimalaiKhareChristou2018" class="citation journal cs1">Dinov, Ivo; 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(November 2012)">full citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text">Barnard, G.A. (1995) "Pivotal Models and the Fiducial Argument", International Statistical Review, 63 (3), 309–323. <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/1403482">1403482</a></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraser1968" class="citation book cs1">Fraser, D. A. S. (1968). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/440926"><i>The structure of inference</i></a>. 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A. S. (1979). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/3559629"><i>Inference and linear models</i></a>. London: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-021910-9" title="Special:BookSources/0-07-021910-9"><bdi>0-07-021910-9</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/3559629">3559629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inference+and+linear+models&rft.place=London&rft.pub=McGraw-Hill&rft.date=1979&rft_id=info%3Aoclcnum%2F3559629&rft.isbn=0-07-021910-9&rft.aulast=Fraser&rft.aufirst=D.+A.+S.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F3559629&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaraldsenLindqvist2013" class="citation journal cs1">Taraldsen, Gunnar; Lindqvist, Bo Henry (2013-02-01). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/annals-of-statistics/volume-41/issue-1/Fiducial-theory-and-optimal-inference/10.1214/13-AOS1083.full">"Fiducial theory and optimal inference"</a>. <i>The Annals of Statistics</i>. <b>41</b> (1). <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/1301.1717">1301.1717</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/10.1214%2F13-AOS1083">10.1214/13-AOS1083</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/0090-5364">0090-5364</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:88520957">88520957</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=Fiducial+theory+and+optimal+inference&rft.volume=41&rft.issue=1&rft.date=2013-02-01&rft_id=info%3Aarxiv%2F1301.1717&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A88520957%23id-name%3DS2CID&rft.issn=0090-5364&rft_id=info%3Adoi%2F10.1214%2F13-AOS1083&rft.aulast=Taraldsen&rft.aufirst=Gunnar&rft.au=Lindqvist%2C+Bo+Henry&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fannals-of-statistics%2Fvolume-41%2Fissue-1%2FFiducial-theory-and-optimal-inference%2F10.1214%2F13-AOS1083.full&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Finetti1937" class="citation journal cs1">De Finetti, Bruno (1937). "La Prévision: ses lois logiques, ses sources subjectives". <i>Annales de l'Institut Henri Poincaré</i>. <b>7</b> (1): 1–68. <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/0365-320X">0365-320X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+l%27Institut+Henri+Poincar%C3%A9&rft.atitle=La+Pr%C3%A9vision%3A+ses+lois+logiques%2C+ses+sources+subjectives&rft.volume=7&rft.issue=1&rft.pages=1-68&rft.date=1937&rft.issn=0365-320X&rft.aulast=De+Finetti&rft.aufirst=Bruno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span> Translated in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Finetti1992" class="citation book cs1">De Finetti, Bruno (1992). "Foresight: Its Logical Laws, Its Subjective Sources". <i>Breakthroughs in Statistics</i>. Springer Series in Statistics. pp. 134–174. <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%2F978-1-4612-0919-5_10">10.1007/978-1-4612-0919-5_10</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94037-3" title="Special:BookSources/978-0-387-94037-3"><bdi>978-0-387-94037-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foresight%3A+Its+Logical+Laws%2C+Its+Subjective+Sources&rft.btitle=Breakthroughs+in+Statistics&rft.series=Springer+Series+in+Statistics&rft.pages=134-174&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0919-5_10&rft.isbn=978-0-387-94037-3&rft.aulast=De+Finetti&rft.aufirst=Bruno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></span> </li> <li id="cite_note-geisser-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-geisser_65-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Seymour_Geisser" title="Seymour Geisser">Geisser, Seymour</a> (1993) <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=wfdlBZ_iwZoC">Predictive Inference: An Introduction</a></i>, CRC Press. <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/0-412-03471-9" title="Special:BookSources/0-412-03471-9">0-412-03471-9</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=28" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBandyopadhyayForster2011" class="citation cs2">Bandyopadhyay, P. S.; Forster, M. R., eds. (2011), <i>Philosophy of Statistics</i>, <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Statistics&rft.pub=Elsevier&rft.date=2011&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBickelDoksum2001" class="citation book cs1">Bickel, Peter J.; Doksum, Kjell A. (2001). <i>Mathematical statistics: Basic and selected topics</i>. Vol. 1 (Second (updated printing 2007) ed.). <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-850363-5" title="Special:BookSources/978-0-13-850363-5"><bdi>978-0-13-850363-5</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0443141">0443141</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%3A+Basic+and+selected+topics&rft.edition=Second+%28updated+printing+2007%29&rft.pub=Prentice+Hall&rft.date=2001&rft.isbn=978-0-13-850363-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D443141%23id-name%3DMR&rft.aulast=Bickel&rft.aufirst=Peter+J.&rft.au=Doksum%2C+Kjell+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><a href="/wiki/David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, D. R.</a> (2006). <i>Principles of Statistical Inference</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <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/0-521-68567-2" title="Special:BookSources/0-521-68567-2">0-521-68567-2</a>.</li> <li><a href="/wiki/Ronald_A._Fisher" class="mw-redirect" title="Ronald A. Fisher">Fisher, R. A.</a> (1955), "Statistical methods and scientific induction", <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society" title="Journal of the Royal Statistical Society">Journal of the Royal Statistical Society</a>, Series B</i>, 17, 69–78. (criticism of statistical theories of <a href="/wiki/Jerzy_Neyman" title="Jerzy Neyman">Jerzy Neyman</a> and <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreedman2009" class="citation book cs1"><a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman, D. A.</a> (2009). <i>Statistical Models: Theory and practice</i> (revised ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. xiv+442 pp. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-74385-3" title="Special:BookSources/978-0-521-74385-3"><bdi>978-0-521-74385-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2489600">2489600</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Models%3A+Theory+and+practice&rft.pages=xiv%2B442+pp&rft.edition=revised&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-74385-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2489600%23id-name%3DMR&rft.aulast=Freedman&rft.aufirst=D.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman, D. A.</a> (2010). <i>Statistical Models and Causal Inferences: A Dialogue with the Social Sciences</i> (Edited by <a href="/wiki/David_Collier_(political_scientist)" title="David Collier (political scientist)">David Collier</a>, <a href="/wiki/Jasjeet_S._Sekhon" title="Jasjeet S. Sekhon">Jasjeet Sekhon</a>, and Philip B. Stark), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHampel2003" class="citation journal cs1">Hampel, Frank R. (February 2003). "The proper fiducial argument". <i>Seminar für Statistik, Eidgenössische Technische Hochschule (ETH)</i>. <b>114</b>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3929%2Fethz-a-004526011">10.3929/ethz-a-004526011</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Seminar+f%C3%BCr+Statistik%2C+Eidgen%C3%B6ssische+Technische+Hochschule+%28ETH%29&rft.atitle=The+proper+fiducial+argument&rft.volume=114&rft.date=2003-02&rft_id=info%3Adoi%2F10.3929%2Fethz-a-004526011&rft.aulast=Hampel&rft.aufirst=Frank+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHansenYu2001" class="citation journal cs1">Hansen, Mark H.; <a href="/wiki/Bin_Yu" title="Bin Yu">Yu, Bin</a> (June 2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041116080440/http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps">"Model Selection and the Principle of Minimum Description Length: Review paper"</a>. <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>96</b> (454): 746–774. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.6581">10.1.1.43.6581</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/10.1198%2F016214501753168398">10.1198/016214501753168398</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/2670311">2670311</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1939352">1939352</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:14460386">14460386</a>. Archived from <a rel="nofollow" class="external text" href="http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps">the original</a> on 2004-11-16.</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=Model+Selection+and+the+Principle+of+Minimum+Description+Length%3A+Review+paper&rft.volume=96&rft.issue=454&rft.pages=746-774&rft.date=2001-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14460386%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1198%2F016214501753168398&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.43.6581%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1939352%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2670311%23id-name%3DJSTOR&rft.aulast=Hansen&rft.aufirst=Mark+H.&rft.au=Yu%2C+Bin&rft_id=http%3A%2F%2Fwww.stat.berkeley.edu%2Fwebmastr%2Fusers%2Fbinyu%2Fps%2Fmdl.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinkelmannKempthorne2008" class="citation book cs1">Hinkelmann, Klaus; <a href="/wiki/Oscar_Kempthorne" title="Oscar Kempthorne">Kempthorne, Oscar</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T3wWj2kVYZgC"><i>Introduction to Experimental Design</i></a> (Second ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72756-9" title="Special:BookSources/978-0-471-72756-9"><bdi>978-0-471-72756-9</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+Experimental+Design&rft.edition=Second&rft.pub=Wiley&rft.date=2008&rft.isbn=978-0-471-72756-9&rft.aulast=Hinkelmann&rft.aufirst=Klaus&rft.au=Kempthorne%2C+Oscar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT3wWj2kVYZgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmogorov1963" class="citation journal cs1"><a href="/wiki/Andrei_N._Kolmogorov" class="mw-redirect" title="Andrei N. Kolmogorov">Kolmogorov, Andrei N.</a> (1963). "On tables of random numbers". <i>Sankhyā Ser. A</i>. <b>25</b>: 369–375. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0178484">0178484</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sankhy%C4%81+Ser.+A&rft.atitle=On+tables+of+random+numbers&rft.volume=25&rft.pages=369-375&rft.date=1963&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D178484%23id-name%3DMR&rft.aulast=Kolmogorov&rft.aufirst=Andrei+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span> Reprinted as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmogorov1998" class="citation journal cs1">Kolmogorov, Andrei N. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0304-3975%2898%2900075-9">"On tables of random numbers"</a>. <i><a href="/wiki/Theoretical_Computer_Science_(journal)" title="Theoretical Computer Science (journal)">Theoretical Computer Science</a></i>. <b>207</b> (2): 387–395. <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%2FS0304-3975%2898%2900075-9">10.1016/S0304-3975(98)00075-9</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1643414">1643414</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=On+tables+of+random+numbers&rft.volume=207&rft.issue=2&rft.pages=387-395&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0304-3975%2898%2900075-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1643414%23id-name%3DMR&rft.aulast=Kolmogorov&rft.aufirst=Andrei+N.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0304-3975%252898%252900075-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li>Konishi S., Kitagawa G. (2008), <i>Information Criteria and Statistical Modeling</i>, Springer.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKruskal1988" class="citation journal cs1"><a href="/wiki/William_Kruskal" title="William Kruskal">Kruskal, William</a> (December 1988). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2290117">"Miracles and statistics: the casual assumption of independence (ASA Presidential Address)"</a>. <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>83</b> (404): 929–940. <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/10.2307%2F2290117">10.2307/2290117</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/2290117">2290117</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=Miracles+and+statistics%3A+the+casual+assumption+of+independence+%28ASA+Presidential+Address%29&rft.volume=83&rft.issue=404&rft.pages=929-940&rft.date=1988-12&rft_id=info%3Adoi%2F10.2307%2F2290117&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2290117%23id-name%3DJSTOR&rft.aulast=Kruskal&rft.aufirst=William&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F2290117&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><a href="/wiki/Lucien_Le_Cam" title="Lucien Le Cam">Le Cam, Lucian</a>. (1986) <i>Asymptotic Methods of Statistical Decision Theory</i>, Springer. <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/0-387-96307-3" title="Special:BookSources/0-387-96307-3">0-387-96307-3</a></li> <li><a href="/wiki/David_S._Moore" title="David S. Moore">Moore, D. S.</a>; McCabe, G. P.; Craig, B. A. (2015), <i>Introduction to the Practice of Statistics</i>, Eighth Edition, Macmillan.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeyman1956" class="citation journal cs1"><a href="/wiki/Jerzy_Neyman" title="Jerzy Neyman">Neyman, Jerzy</a> (1956). "Note on an article by Sir Ronald Fisher". <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>18</b> (2): 288–294. <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.2517-6161.1956.tb00236.x">10.1111/j.2517-6161.1956.tb00236.x</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/2983716">2983716</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=Note+on+an+article+by+Sir+Ronald+Fisher&rft.volume=18&rft.issue=2&rft.pages=288-294&rft.date=1956&rft_id=info%3Adoi%2F10.1111%2Fj.2517-6161.1956.tb00236.x&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2983716%23id-name%3DJSTOR&rft.aulast=Neyman&rft.aufirst=Jerzy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span> (reply to Fisher 1955)</li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (1877–1878), "Illustrations of the logic of science" (series), <i><a href="/wiki/Popular_Science_Monthly" class="mw-redirect" title="Popular Science Monthly">Popular Science Monthly</a></i>, vols. 12–13. Relevant individual papers: <ul><li>(1878 March), "The Doctrine of Chances", <i>Popular Science Monthly</i>, v. 12, March issue, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZKMVAAAAYAAJ">604</a>–615. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popscimonthly12yoummiss#page/612/mode/1up">Eprint</a>.</li> <li>(1878 April), "The Probability of Induction", <i>Popular Science Monthly</i>, v. 12, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZKMVAAAAYAAJ">705</a>–718. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popscimonthly12yoummiss#page/715/mode/1up">Eprint</a>.</li> <li>(1878 June), "The Order of Nature", <i>Popular Science Monthly</i>, v. 13, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u8sWAQAAIAAJ">203</a>–217.<i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popularsciencemo13newy#page/203/mode/1up">Eprint</a>.</li> <li>(1878 August), "Deduction, Induction, and Hypothesis", <i>Popular Science Monthly</i>, v. 13, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u8sWAQAAIAAJ">470</a>–482. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popularsciencemo13newy#page/470/mode/1up">Eprint</a>.</li></ul></li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (1883), "A Theory of probable inference", <i>Studies in Logic</i>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_V7oIAAAAQAAJ/page/n134">126-181</a>, Little, Brown, and Company. (Reprinted 1983, <a href="/wiki/John_Benjamins_Publishing_Company" title="John Benjamins Publishing Company">John Benjamins Publishing Company</a>, <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/90-272-3271-7" title="Special:BookSources/90-272-3271-7">90-272-3271-7</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreedmanPisaniPurves1978" class="citation book cs1"><a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman, D.A</a>; Pisani, R.; Purves, R.A. (1978). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/statistics0000free"><i>Statistics</i></a></span>. New York: <a href="/w/index.php?title=W.W._Norton,_Inc.&action=edit&redlink=1" class="new" title="W.W. Norton, Inc. (page does not exist)">W. W. Norton & Company</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics&rft.place=New+York&rft.pub=W.+W.+Norton+%26+Company&rft.date=1978&rft.aulast=Freedman&rft.aufirst=D.A&rft.au=Pisani%2C+R.&rft.au=Purves%2C+R.A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstatistics0000free&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfanzaglwith_the_assistance_of_R._Hamböker1994" class="citation book cs1">Pfanzagl, Johann; with the assistance of R. Hamböker (1994). <i>Parametric Statistical Theory</i>. Berlin: <a href="/wiki/Walter_de_Gruyter" class="mw-redirect" title="Walter de Gruyter">Walter de Gruyter</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-013863-4" title="Special:BookSources/978-3-11-013863-4"><bdi>978-3-11-013863-4</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1291393">1291393</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parametric+Statistical+Theory&rft.place=Berlin&rft.pub=Walter+de+Gruyter&rft.date=1994&rft.isbn=978-3-11-013863-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1291393%23id-name%3DMR&rft.aulast=Pfanzagl&rft.aufirst=Johann&rft.au=with+the+assistance+of+R.+Hamb%C3%B6ker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRissanen1989" class="citation book cs1"><a href="/wiki/Jorma_Rissanen" title="Jorma Rissanen">Rissanen, Jorma</a> (1989). <i>Stochastic Complexity in Statistical Inquiry</i>. Series in Computer Science. Vol. 15. Singapore: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9971-5-0859-3" title="Special:BookSources/978-9971-5-0859-3"><bdi>978-9971-5-0859-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1082556">1082556</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Complexity+in+Statistical+Inquiry&rft.place=Singapore&rft.series=Series+in+Computer+Science&rft.pub=World+Scientific&rft.date=1989&rft.isbn=978-9971-5-0859-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1082556%23id-name%3DMR&rft.aulast=Rissanen&rft.aufirst=Jorma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoofi2000" class="citation journal cs1">Soofi, Ehsan S. (December 2000). "Principal information-theoretic approaches (Vignettes for the Year 2000: Theory and Methods, ed. by George Casella)". <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>95</b> (452): 1349–1353. <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%2F01621459.2000.10474346">10.1080/01621459.2000.10474346</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/2669786">2669786</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1825292">1825292</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:120143121">120143121</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=Principal+information-theoretic+approaches+%28Vignettes+for+the+Year+2000%3A+Theory+and+Methods%2C+ed.+by+George+Casella%29&rft.volume=95&rft.issue=452&rft.pages=1349-1353&rft.date=2000-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120143121%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1825292%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2669786%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F01621459.2000.10474346&rft.aulast=Soofi&rft.aufirst=Ehsan+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTraubWasilkowskiWozniakowski1988" class="citation book cs1"><a href="/wiki/Joseph_F._Traub" title="Joseph F. Traub">Traub, Joseph F.</a>; Wasilkowski, G. W.; Wozniakowski, H. (1988). <i>Information-Based Complexity</i>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-697545-1" title="Special:BookSources/978-0-12-697545-1"><bdi>978-0-12-697545-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information-Based+Complexity&rft.pub=Academic+Press&rft.date=1988&rft.isbn=978-0-12-697545-1&rft.aulast=Traub&rft.aufirst=Joseph+F.&rft.au=Wasilkowski%2C+G.+W.&rft.au=Wozniakowski%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZabell1992" class="citation journal cs1">Zabell, S. L. (Aug 1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177011233">"R. A. Fisher and Fiducial Argument"</a>. <i>Statistical Science</i>. <b>7</b> (3): 369–387. <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%2F1177011233">10.1214/ss/1177011233</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/2246073">2246073</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=R.+A.+Fisher+and+Fiducial+Argument&rft.volume=7&rft.issue=3&rft.pages=369-387&rft.date=1992-08&rft_id=info%3Adoi%2F10.1214%2Fss%2F1177011233&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2246073%23id-name%3DJSTOR&rft.aulast=Zabell&rft.aufirst=S.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Fss%252F1177011233&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li></ul> </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=Statistical_inference&action=edit&section=29" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/George_Casella" title="George Casella">Casella, G.</a>, <a href="/wiki/Roger_Lee_Berger" title="Roger Lee Berger">Berger, R. L.</a> (2002). <i>Statistical Inference</i>. Duxbury Press. <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/0-534-24312-6" title="Special:BookSources/0-534-24312-6">0-534-24312-6</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreedman1991" class="citation journal cs1"><a href="/wiki/David_A._Freedman" title="David A. Freedman">Freedman, D.A.</a> (1991). "Statistical models and shoe leather". <i>Sociological Methodology</i>. <b>21</b>: 291–313. <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%2F270939">10.2307/270939</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/270939">270939</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sociological+Methodology&rft.atitle=Statistical+models+and+shoe+leather&rft.volume=21&rft.pages=291-313&rft.date=1991&rft_id=info%3Adoi%2F10.2307%2F270939&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F270939%23id-name%3DJSTOR&rft.aulast=Freedman&rft.aufirst=D.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li>Held L., Bové D.S. (2014). <i>Applied Statistical Inference—Likelihood and Bayes</i> (Springer).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLenhard2006" class="citation journal cs1">Lenhard, Johannes (2006). <a rel="nofollow" class="external text" href="http://www.stats.org.uk/statistical-inference/Lenhard2006.pdf">"Models and Statistical Inference: the controversy between Fisher and Neyman–Pearson"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/British_Journal_for_the_Philosophy_of_Science" title="British Journal for the Philosophy of Science">British Journal for the Philosophy of Science</a></i>. <b>57</b>: 69–91. <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%2Fbjps%2Faxi152">10.1093/bjps/axi152</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:14136146">14136146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=British+Journal+for+the+Philosophy+of+Science&rft.atitle=Models+and+Statistical+Inference%3A+the+controversy+between+Fisher+and+Neyman%E2%80%93Pearson&rft.volume=57&rft.pages=69-91&rft.date=2006&rft_id=info%3Adoi%2F10.1093%2Fbjps%2Faxi152&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14136146%23id-name%3DS2CID&rft.aulast=Lenhard&rft.aufirst=Johannes&rft_id=http%3A%2F%2Fwww.stats.org.uk%2Fstatistical-inference%2FLenhard2006.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLindley1958" class="citation journal cs1">Lindley, D (1958). "Fiducial distribution and Bayes' theorem". <i>Journal of the Royal Statistical Society, Series B</i>. <b>20</b>: 102–7. <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.2517-6161.1958.tb00278.x">10.1111/j.2517-6161.1958.tb00278.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=Fiducial+distribution+and+Bayes%27+theorem&rft.volume=20&rft.pages=102-7&rft.date=1958&rft_id=info%3Adoi%2F10.1111%2Fj.2517-6161.1958.tb00278.x&rft.aulast=Lindley&rft.aufirst=D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li>Rahlf, Thomas (2014). "Statistical Inference", in Claude Diebolt, and Michael Haupert (eds.), "Handbook of Cliometrics ( Springer Reference Series)", Berlin/Heidelberg: Springer.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReidCox2014" class="citation journal cs1">Reid, N.; Cox, D. R. (2014). "On Some Principles of Statistical Inference". <i>International Statistical Review</i>. <b>83</b> (2): 293–308. <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%2Finsr.12067">10.1111/insr.12067</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/10.1111%2Finsr.12067">10.1111/insr.12067</a></span>. <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:17410547">17410547</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review&rft.atitle=On+Some+Principles+of+Statistical+Inference&rft.volume=83&rft.issue=2&rft.pages=293-308&rft.date=2014&rft_id=info%3Ahdl%2F10.1111%2Finsr.12067&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17410547%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Finsr.12067&rft.aulast=Reid&rft.aufirst=N.&rft.au=Cox%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStatistical+inference" class="Z3988"></span></li> <li>Sagitov, Serik (2022). "Statistical Inference". Wikibooks. <a class="external free" href="https://upload.wikimedia.org/wikipedia/commons/f/f9/Statistical_Inference.pdf">http://upload.wikimedia.org/wikipedia/commons/f/f9/Statistical_Inference.pdf</a></li> <li>Young, G.A., Smith, R.L. (2005). <i>Essentials of Statistical Inference</i>, CUP. <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/0-521-83971-8" title="Special:BookSources/0-521-83971-8">0-521-83971-8</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Statistical_inference&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output 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has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Statistical_inference" class="extiw" title="commons:Category:Statistical inference">Statistical inference</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, 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Learning</a></li> <li>An online, Bayesian (MCMC) demo/calculator is available at <a rel="nofollow" class="external text" href="https://causascientia.org">causaScientia</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output 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dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a 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 class="mw-selflink selflink">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 href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">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 href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">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 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