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class="vector-toc-link" href="#Estimation_of_location"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Estimation of location</span> </div> </a> <ul id="toc-Estimation_of_location-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estimation_of_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation_of_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Estimation of scale</span> </div> </a> <ul id="toc-Estimation_of_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Manual_screening_for_outliers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Manual_screening_for_outliers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Manual screening for outliers</span> </div> </a> <ul id="toc-Manual_screening_for_outliers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variety_of_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variety_of_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Variety of applications</span> </div> </a> <ul id="toc-Variety_of_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measures_of_robustness" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Measures_of_robustness"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Measures of robustness</span> </div> </a> <button aria-controls="toc-Measures_of_robustness-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 Measures of robustness subsection</span> </button> <ul id="toc-Measures_of_robustness-sublist" class="vector-toc-list"> <li id="toc-Breakdown_point" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Breakdown_point"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Breakdown point</span> </div> </a> <ul id="toc-Breakdown_point-sublist" class="vector-toc-list"> <li id="toc-Example:_speed-of-light_data" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example:_speed-of-light_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Example: speed-of-light data</span> </div> </a> <ul id="toc-Example:_speed-of-light_data-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Empirical_influence_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Empirical_influence_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Empirical influence function</span> </div> </a> <ul id="toc-Empirical_influence_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence_function_and_sensitivity_curve" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Influence_function_and_sensitivity_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Influence function and sensitivity curve</span> </div> </a> <ul id="toc-Influence_function_and_sensitivity_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Desirable_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Desirable_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Desirable properties</span> </div> </a> <ul id="toc-Desirable_properties-sublist" class="vector-toc-list"> <li id="toc-Rejection_point" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rejection_point"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Rejection point</span> </div> </a> <ul id="toc-Rejection_point-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gross-error_sensitivity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gross-error_sensitivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.2</span> <span>Gross-error sensitivity</span> </div> </a> <ul id="toc-Gross-error_sensitivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local-shift_sensitivity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Local-shift_sensitivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.3</span> <span>Local-shift sensitivity</span> </div> </a> <ul id="toc-Local-shift_sensitivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-M-estimators" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#M-estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>M-estimators</span> </div> </a> <button aria-controls="toc-M-estimators-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 M-estimators subsection</span> </button> <ul id="toc-M-estimators-sublist" class="vector-toc-list"> <li id="toc-Properties_of_M-estimators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_of_M-estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Properties of M-estimators</span> </div> </a> <ul id="toc-Properties_of_M-estimators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence_function_of_an_M-estimator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Influence_function_of_an_M-estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Influence function of an M-estimator</span> </div> </a> <ul id="toc-Influence_function_of_an_M-estimator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Choice_of_ψ_and_ρ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Choice_of_ψ_and_ρ"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Choice of <i>ψ</i> and <i>ρ</i></span> </div> </a> <ul id="toc-Choice_of_ψ_and_ρ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Robust_parametric_approaches" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Robust_parametric_approaches"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Robust parametric approaches</span> </div> </a> <button aria-controls="toc-Robust_parametric_approaches-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 Robust parametric approaches subsection</span> </button> <ul id="toc-Robust_parametric_approaches-sublist" class="vector-toc-list"> <li id="toc-Example:_speed-of-light_data_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_speed-of-light_data_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Example: speed-of-light data</span> </div> </a> <ul id="toc-Example:_speed-of-light_data_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related concepts</span> </div> </a> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Replacing_outliers_and_missing_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Replacing_outliers_and_missing_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Replacing outliers and missing values</span> </div> </a> <ul id="toc-Replacing_outliers_and_missing_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Use_in_machine_learning" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Use_in_machine_learning"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Use in machine learning</span> </div> </a> <ul id="toc-Use_in_machine_learning-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">10</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">11</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">12</span> <span>References</span> </div> </a> <ul id="toc-References-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">13</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" 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statistics</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://ca.wikipedia.org/wiki/Estad%C3%ADstica_robusta" title="Estadística robusta – Catalan" lang="ca" hreflang="ca" data-title="Estadística robusta" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Robuste_Sch%C3%A4tzverfahren" title="Robuste Schätzverfahren – German" lang="de" hreflang="de" data-title="Robuste Schätzverfahren" 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_robusta" title="Estadística robusta – Spanish" lang="es" hreflang="es" data-title="Estadística robusta" 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/Sendotasun_(estatistika)" title="Sendotasun (estatistika) – Basque" lang="eu" hreflang="eu" data-title="Sendotasun (estatistika)" 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%A2%D9%85%D8%A7%D8%B1_%D9%85%D9%82%D8%A7%D9%88%D9%85" title="آمار مقاوم – Persian" lang="fa" hreflang="fa" data-title="آمار مقاوم" 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/Robustesse_(statistiques)" title="Robustesse (statistiques) – French" lang="fr" hreflang="fr" data-title="Robustesse (statistiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94_%D7%97%D7%A1%D7%99%D7%A0%D7%94" title="סטטיסטיקה חסינה – Hebrew" lang="he" hreflang="he" data-title="סטטיסטיקה חסינה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Statystyka_odporno%C5%9Bciowa" title="Statystyka odpornościowa – Polish" lang="pl" hreflang="pl" data-title="Statystyka odpornościowa" 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/Estat%C3%ADstica_robusta" title="Estatística robusta – Portuguese" lang="pt" hreflang="pt" data-title="Estatística robusta" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Robust_statistic" title="Robust statistic – Romanian" lang="ro" hreflang="ro" data-title="Robust statistic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B1%D0%B0%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Робастность – Russian" lang="ru" hreflang="ru" data-title="Робастность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B1%D0%B0%D1%81%D1%82%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D1%83_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%86%D1%96" title="Робастність у статистиці – Ukrainian" lang="uk" hreflang="uk" data-title="Робастність у статистиці" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%A0%91%E5%81%A5%E7%B5%B1%E8%A8%88" title="頑健統計 – Cantonese" lang="yue" hreflang="yue" data-title="頑健統計" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A8%B3%E5%81%A5%E7%BB%9F%E8%AE%A1" title="稳健统计 – Chinese" lang="zh" hreflang="zh" data-title="稳健统计" data-language-autonym="中文" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of statistics</div> <p><b>Robust statistics</b> are <a href="/wiki/Statistics" title="Statistics">statistics</a> that maintain their properties even if the underlying distributional assumptions are incorrect. Robust <a href="/wiki/Statistics" title="Statistics">statistical</a> methods have been developed for many common problems, such as estimating <a href="/wiki/Location_parameter" title="Location parameter">location</a>, <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a>, and <a href="/wiki/Regression_coefficient" class="mw-redirect" title="Regression coefficient">regression parameters</a>. One motivation is to produce <a href="/wiki/Statistical_method" class="mw-redirect" title="Statistical method">statistical methods</a> that are not unduly affected by <a href="/wiki/Outlier" title="Outlier">outliers</a>. Another motivation is to provide methods with good performance when there are small departures from a <a href="/wiki/Parametric_statistics" title="Parametric statistics">parametric distribution</a>. For example, robust methods work well for mixtures of two <a href="/wiki/Normal_distribution" title="Normal distribution">normal distributions</a> with different <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a>; under this model, non-robust methods like a <a href="/wiki/T-test" class="mw-redirect" title="T-test">t-test</a> work poorly.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> </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=Robust_statistics&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from <a href="/wiki/Statistical_assumption" title="Statistical assumption">model assumptions</a>. In statistics, classical estimation methods rely heavily on assumptions that are often not met in practice. In particular, it is often assumed that the data errors are normally distributed, at least approximately, or that the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> can be relied on to produce normally distributed estimates. Unfortunately, when there are outliers in the data, classical <a href="/wiki/Estimator" title="Estimator">estimators</a> often have very poor performance, when judged using the <i><a href="#Breakdown_point">breakdown point</a></i> and the <i><a href="#Influence_function_and_sensitivity_curve">influence function</a></i> described below. </p><p>The practical effect of problems seen in the influence function can be studied empirically by examining the <a href="/wiki/Sampling_distribution" title="Sampling distribution">sampling distribution</a> of proposed estimators under a <a href="/wiki/Mixture_model" title="Mixture model">mixture model</a>, where one mixes in a small amount (1–5% is often sufficient) of contamination. For instance, one may use a mixture of 95% a normal distribution, and 5% a normal distribution with the same mean but significantly higher standard deviation (representing outliers). </p><p>Robust <a href="/wiki/Parametric_statistics" title="Parametric statistics">parametric statistics</a> can proceed in two ways: </p> <ul><li>by designing estimators so that a pre-selected behaviour of the influence function is achieved</li> <li>by replacing estimators that are optimal under the assumption of a normal distribution with estimators that are optimal for, or at least derived for, other distributions; for example, using the <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution"><i>t</i>-distribution</a> with low degrees of freedom (high kurtosis) or with a <a href="/wiki/Mixture_density" class="mw-redirect" title="Mixture density">mixture</a> of two or more distributions.</li></ul> <p>Robust estimates have been studied for the following problems: </p> <ul><li>estimating <a href="/wiki/Location_parameter" title="Location parameter">location parameters</a></li> <li>estimating <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameters</a></li> <li>estimating <a href="/wiki/Regression_coefficient" class="mw-redirect" title="Regression coefficient">regression coefficients</a><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></li> <li>estimation of model-states in models expressed in <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state-space</a> form, for which the standard method is equivalent to a <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-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=Robust_statistics&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">July 2008</span>)</i></span></div></td></tr></tbody></table> <p>There are various definitions of a "robust <a href="/wiki/Statistic" title="Statistic">statistic</a>". Strictly speaking, a <b>robust <a href="/wiki/Statistic" title="Statistic">statistic</a></b> is resistant to errors in the results, produced by deviations from assumptions<sup id="cite_ref-huber_4-0" class="reference"><a href="#cite_note-huber-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (e.g., of normality). This means that if the assumptions are only approximately met, the <b>robust estimator</b> will still have a reasonable <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficiency</a>, and reasonably small <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a>, as well as being <a href="/wiki/Asymptotically_unbiased" class="mw-redirect" title="Asymptotically unbiased">asymptotically unbiased</a>, meaning having a bias tending towards 0 as the sample size tends towards infinity. </p><p>Usually, the most important case is <b>distributional robustness</b> - robustness to breaking of the assumptions about the underlying distribution of the data.<sup id="cite_ref-huber_4-1" class="reference"><a href="#cite_note-huber-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Classical statistical procedures are typically sensitive to "longtailedness" (e.g., when the distribution of the data has longer tails than the assumed normal distribution). This implies that they will be strongly affected by the presence of <a href="/wiki/Outlier" title="Outlier">outliers</a> in the data, and the estimates they produce may be heavily distorted if there are extreme outliers in the data, compared to what they would be if the outliers were not included in the data. </p><p>By contrast, more robust estimators that are not so sensitive to distributional distortions such as longtailedness are also resistant to the presence of outliers. Thus, in the context of robust statistics, <i>distributionally robust</i> and <i>outlier-resistant</i> are effectively synonymous.<sup id="cite_ref-huber_4-2" class="reference"><a href="#cite_note-huber-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For one perspective on research in robust statistics up to 2000, see <a href="#CITEREFPortnoyHe2000">Portnoy & He (2000)</a>. </p><p>Some experts prefer the term <b>resistant statistics</b> for distributional robustness, and reserve 'robustness' for non-distributional robustness, e.g., robustness to violation of assumptions about the probability model or estimator, but this is a minority usage. Plain 'robustness' to mean 'distributional robustness' is common. </p><p>When considering how robust an estimator is to the presence of outliers, it is useful to test what happens when an extreme outlier is <b>added</b> to the dataset, and to test what happens when an extreme outlier <b>replaces</b> one of the existing data points, and then to consider the effect of multiple additions or replacements. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean</a> is not a robust measure of <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a>. If the dataset is, e.g., the values {2,3,5,6,9}, then if we add another datapoint with value -1000 or +1000 to the data, the resulting mean will be very different from the mean of the original data. Similarly, if we replace one of the values with a datapoint of value -1000 or +1000 then the resulting mean will be very different from the mean of the original data. </p><p>The <a href="/wiki/Median" title="Median">median</a> is a robust measure of <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a>. Taking the same dataset {2,3,5,6,9}, if we add another datapoint with value -1000 or +1000 then the median will change slightly, but it will still be similar to the median of the original data. If we replace one of the values with a data point of value -1000 or +1000 then the resulting median will still be similar to the median of the original data. </p><p>Described in terms of <a href="#Breakdown_point">breakdown points</a>, the median has a breakdown point of 50%, meaning that half the points must be outliers before the median can be moved outside the range of the non-outliers, while the mean has a breakdown point of 0, as a single large observation can throw it off. </p><p>The <a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">median absolute deviation</a> and <a href="/wiki/Interquartile_range" title="Interquartile range">interquartile range</a> are robust measures of <a href="/wiki/Statistical_dispersion" title="Statistical dispersion">statistical dispersion</a>, while the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> and <a href="/wiki/Range_(statistics)" title="Range (statistics)">range</a> are not. </p><p><a href="/wiki/Trimmed_estimator" title="Trimmed estimator">Trimmed estimators</a> and <a href="/wiki/Winsorising" class="mw-redirect" title="Winsorising">Winsorised estimators</a> are general methods to make statistics more robust. <a href="/wiki/L-estimator" title="L-estimator">L-estimators</a> are a general class of simple statistics, often robust, while <a href="#M-estimators">M-estimators</a> are a general class of robust statistics, and are now the preferred solution, though they can be quite involved to calculate. </p> <div class="mw-heading mw-heading3"><h3 id="Speed-of-light_data">Speed-of-light data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=4" title="Edit section: Speed-of-light data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Andrew_Gelman" title="Andrew Gelman">Gelman</a> et al. in Bayesian Data Analysis (2004) consider a data set relating to <a href="/wiki/Speed-of-light" class="mw-redirect" title="Speed-of-light">speed-of-light</a> measurements made by <a href="/wiki/Simon_Newcomb" title="Simon Newcomb">Simon Newcomb</a>. The data sets for that book can be found via the <a href="/wiki/Classic_data_sets" class="mw-redirect" title="Classic data sets">Classic data sets</a> page, and the book's website contains more information on the data. </p><p>Although the bulk of the data looks to be more or less normally distributed, there are two obvious outliers. These outliers have a large effect on the mean, dragging it towards them, and away from the center of the bulk of the data. Thus, if the mean is intended as a measure of the location of the center of the data, it is, in a sense, biased when outliers are present. </p><p>Also, the distribution of the mean is known to be asymptotically normal due to the central limit theorem. However, outliers can make the distribution of the mean non-normal, even for fairly large data sets. Besides this non-normality, the mean is also <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">inefficient</a> in the presence of outliers and less variable measures of location are available. </p> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_location">Estimation of location</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=5" title="Edit section: Estimation of location"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The plot below shows a density plot of the speed-of-light data, together with a <a href="/wiki/Rug_plot" title="Rug plot">rug plot</a> (panel (a)). Also shown is a normal <a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a> (panel (b)). The outliers are visible in these plots. </p><p>Panels (c) and (d) of the plot show the bootstrap distribution of the mean (c) and the 10% <a href="/wiki/Trimmed_mean" class="mw-redirect" title="Trimmed mean">trimmed mean</a> (d). The trimmed mean is a simple, robust estimator of location that deletes a certain percentage of observations (10% here) <i>from each end</i> of the data, then computes the mean in the usual way. The analysis was performed in <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> and 10,000 <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrap</a> samples were used for each of the raw and trimmed means. </p><p>The distribution of the mean is clearly much wider than that of the 10% trimmed mean (the plots are on the same scale). Also whereas the distribution of the trimmed mean appears to be close to normal, the distribution of the raw mean is quite skewed to the left. So, in this sample of 66 observations, only 2 outliers cause the central limit theorem to be inapplicable. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:SpeedOfLight.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e6/SpeedOfLight.png" decoding="async" width="406" height="306" class="mw-file-element" data-file-width="406" data-file-height="306" /></a></span> </p><p>Robust statistical methods, of which the trimmed mean is a simple example, seek to outperform classical statistical methods in the presence of outliers, or, more generally, when underlying parametric assumptions are not quite correct. </p><p>Whilst the trimmed mean performs well relative to the mean in this example, better robust estimates are available. In fact, the mean, median and trimmed mean are all special cases of <a href="/wiki/M-estimators" class="mw-redirect" title="M-estimators">M-estimators</a>. Details appear in the sections below. </p> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_scale">Estimation of scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=6" title="Edit section: Estimation of scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Robust_measures_of_scale" title="Robust measures of scale">Robust measures of scale</a></div> <p>The outliers in the speed-of-light data have more than just an adverse effect on the mean; the usual estimate of scale is the standard deviation, and this quantity is even more badly affected by outliers because the squares of the deviations from the mean go into the calculation, so the outliers' effects are exacerbated. </p><p>The plots below show the bootstrap distributions of the standard deviation, the <a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">median absolute deviation</a> (MAD) and the Rousseeuw–Croux <a href="/wiki/Robust_measures_of_scale#Robust_measures_of_scale_based_on_absolute_pairwise_differences" title="Robust measures of scale">(Qn) estimator</a> of scale.<sup id="cite_ref-FOOTNOTERousseeuwCroux1993_5-0" class="reference"><a href="#cite_note-FOOTNOTERousseeuwCroux1993-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The plots are based on 10,000 bootstrap samples for each estimator, with some Gaussian noise added to the resampled data (<a href="/wiki/Bootstrapping_(statistics)#Smooth_bootstrap" title="Bootstrapping (statistics)">smoothed bootstrap</a>). Panel (a) shows the distribution of the standard deviation, (b) of the MAD and (c) of Qn. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:SpeedOfLightScale.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a6/SpeedOfLightScale.png" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="400" data-file-height="300" /></a></span> </p><p>The distribution of standard deviation is erratic and wide, a result of the outliers. The MAD is better behaved, and Qn is a little bit more efficient than MAD. This simple example demonstrates that when outliers are present, the standard deviation cannot be recommended as an estimate of scale. </p> <div class="mw-heading mw-heading3"><h3 id="Manual_screening_for_outliers">Manual screening for outliers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=7" title="Edit section: Manual screening for outliers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Traditionally, statisticians would manually screen data for <a href="/wiki/Outliers" class="mw-redirect" title="Outliers">outliers</a>, and remove them, usually checking the source of the data to see whether the outliers were erroneously recorded. Indeed, in the speed-of-light example above, it is easy to see and remove the two outliers prior to proceeding with any further analysis. However, in modern times, data sets often consist of large numbers of variables being measured on large numbers of experimental units. Therefore, manual screening for outliers is often impractical. </p><p>Outliers can often interact in such a way that they mask each other. As a simple example, consider a small univariate data set containing one modest and one large outlier. The estimated standard deviation will be grossly inflated by the large outlier. The result is that the modest outlier looks relatively normal. As soon as the large outlier is removed, the estimated standard deviation shrinks, and the modest outlier now looks unusual. </p><p>This problem of masking gets worse as the complexity of the data increases. For example, in <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> problems, diagnostic plots are used to identify outliers. However, it is common that once a few outliers have been removed, others become visible. The problem is even worse in higher dimensions. </p><p>Robust methods provide automatic ways of detecting, downweighting (or removing), and flagging outliers, largely removing the need for manual screening. Care must be taken; initial data showing the <a href="/wiki/Ozone_hole" class="mw-redirect" title="Ozone hole">ozone hole</a> first appearing over <a href="/wiki/Antarctica" title="Antarctica">Antarctica</a> were rejected as outliers by non-human screening.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Variety_of_applications">Variety of applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=8" title="Edit section: Variety of applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although this article deals with general principles for univariate statistical methods, robust methods also exist for regression problems, generalized linear models, and parameter estimation of various distributions. </p> <div class="mw-heading mw-heading2"><h2 id="Measures_of_robustness">Measures of robustness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=9" title="Edit section: Measures of robustness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic tools used to describe and measure robustness are the <i>breakdown point</i>, the <i>influence function</i> and the <i>sensitivity curve</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Breakdown_point">Breakdown point</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=10" title="Edit section: Breakdown point"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Intuitively, the breakdown point of an <a href="/wiki/Estimator" title="Estimator">estimator</a> is the proportion of incorrect observations (e.g. arbitrarily large observations) an estimator can handle before giving an incorrect (e.g., arbitrarily large) result. Usually, the asymptotic (infinite sample) limit is quoted as the breakdown point, although the finite-sample breakdown point may be more useful.<sup id="cite_ref-maronna_7-0" class="reference"><a href="#cite_note-maronna-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, given <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> independent random 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},\dots ,X_{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> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},\dots ,X_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c90af2f9c5e6071c941c23f72c69f5de62af4b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.109ex; height:2.843ex;" alt="{\displaystyle (X_{1},\dots ,X_{n})}"></span> and the corresponding realizations <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},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}"></span>, we can use <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 {\overline {X_{n}}}:={\frac {X_{1}+\cdots +X_{n}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X_{n}}}:={\frac {X_{1}+\cdots +X_{n}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc10907a821394dad9a7f898a8566b85b31d4c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.364ex; height:5.176ex;" alt="{\displaystyle {\overline {X_{n}}}:={\frac {X_{1}+\cdots +X_{n}}{n}}}"></span> to estimate the mean. Such an estimator has a breakdown point of 0 (or finite-sample breakdown point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span>) because we can make <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 {\overline {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"></span> arbitrarily large just by changing any of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}"></span>. </p><p>The higher the breakdown point of an estimator, the more robust it is. Intuitively, we can understand that a breakdown point cannot exceed 50% because if more than half of the observations are contaminated, it is not possible to distinguish between the underlying distribution and the contaminating distribution <a href="#CITEREFRousseeuwLeroy1987">Rousseeuw & Leroy (1987)</a>. Therefore, the maximum breakdown point is 0.5 and there are estimators which achieve such a breakdown point. For example, the median has a breakdown point of 0.5. The X% trimmed mean has a breakdown point of X%, for the chosen level of X. <a href="#CITEREFHuber1981">Huber (1981)</a> and <a href="#CITEREFMaronnaMartinYohaiSalibián-Barrera2019">Maronna et al. (2019)</a> contain more details. The level and the power breakdown points of tests are investigated in <a href="#CITEREFHeSimpsonPortnoy1990">He, Simpson & Portnoy (1990)</a>. </p><p><span id="resistant_statistic"></span>Statistics with high breakdown points are sometimes called <b>resistant statistics.</b><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Example:_speed-of-light_data">Example: speed-of-light data</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=11" title="Edit section: Example: speed-of-light data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the speed-of-light example, removing the two lowest observations causes the mean to change from 26.2 to 27.75, a change of 1.55. The estimate of scale produced by the Qn method is 6.3. We can divide this by the square root of the sample size to get a robust standard error, and we find this quantity to be 0.78. Thus, the change in the mean resulting from removing two <a href="/wiki/Outliers" class="mw-redirect" title="Outliers">outliers</a> is approximately twice the robust standard error. </p><p>The 10% trimmed mean for the speed-of-light data is 27.43. Removing the two lowest observations and recomputing gives 27.67. The trimmed mean is less affected by the outliers and has a higher breakdown point. </p><p>If we replace the lowest observation, −44, by −1000, the mean becomes 11.73, whereas the 10% trimmed mean is still 27.43. In many areas of applied statistics, it is common for data to be log-transformed to make them near symmetrical. Very small values become large negative when log-transformed, and zeroes become negatively infinite. Therefore, this example is of practical interest. </p> <div class="mw-heading mw-heading3"><h3 id="Empirical_influence_function">Empirical influence function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=12" title="Edit section: Empirical influence function"><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/Cook%27s_distance" title="Cook's distance">Cook's distance</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Robust_statistics&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">June 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Robust_statistics" title="Special:EditPage/Robust statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The empirical influence function is a measure of the dependence of the estimator on the value of any one of the points in the sample. It is a model-free measure in the sense that it simply relies on calculating the estimator again with a different sample. On the right is Tukey's biweight function, which, as we will later see, is an example of what a "good" (in a sense defined later on) empirical influence function should look like. </p><p>In mathematical terms, an influence function is defined as a vector in the space of the estimator, which is in turn defined for a sample which is a subset of the population: </p> <ol><li><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 (\Omega ,{\mathcal {A}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {A}},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc7f54d9e2ab1101bd16dcac877cce9860ae209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},P)}"></span> is a probability space,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {X}},\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {X}},\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf554f3b24128e2486c94287d6611f9d7797843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {X}},\Sigma )}"></span> is a measurable space (state space),</li> <li><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 mathvariant="normal">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/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }"></span> is a <a href="/wiki/Parameter_space" title="Parameter space">parameter space</a> of dimension <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\in \mathbb {N} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {N} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a61840ab9791ffa8925e4e61bcae24edb41741c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.832ex; height:2.676ex;" alt="{\displaystyle p\in \mathbb {N} ^{*}}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma ,S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Gamma ,S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a729125bb6ef651601ebedb09feb405b292e6bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.795ex; height:2.843ex;" alt="{\displaystyle (\Gamma ,S)}"></span> is a measurable space,</li></ol> <p>For example, </p> <ol><li><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 (\Omega ,{\mathcal {A}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {A}},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc7f54d9e2ab1101bd16dcac877cce9860ae209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},P)}"></span> is any probability space,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {X}},\Sigma )=(\mathbb {R} ,{\mathcal {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {X}},\Sigma )=(\mathbb {R} ,{\mathcal {B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec88995372b35d44dfc21203e88741b0ad278f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.56ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {X}},\Sigma )=(\mathbb {R} ,{\mathcal {B}})}"></span>,</li> <li><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 =\mathbb {R} \times \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta =\mathbb {R} \times \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54297285ea56999be16abdc4459238650202e9fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.614ex; height:2.509ex;" alt="{\displaystyle \Theta =\mathbb {R} \times \mathbb {R} ^{+}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma ,S)=(\mathbb {R} ,{\mathcal {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Gamma ,S)=(\mathbb {R} ,{\mathcal {B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1c826fd2fc18fa4493f4650e2b0a5fe7cf7a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.958ex; height:2.843ex;" alt="{\displaystyle (\Gamma ,S)=(\mathbb {R} ,{\mathcal {B}})}"></span>,</li></ol> <p>The empirical influence function is defined as follows. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3c5c71f1af21f3fd4f9d8e1b0c8691ba4258f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.343ex;" alt="{\displaystyle n\in \mathbb {N} ^{*}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,X_{n}:(\Omega ,{\mathcal {A}})\rightarrow ({\mathcal {X}},\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n}:(\Omega ,{\mathcal {A}})\rightarrow ({\mathcal {X}},\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600fed045b755894f2b01d0c8cd7f3df6737223e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.672ex; height:2.843ex;" alt="{\displaystyle X_{1},\dots ,X_{n}:(\Omega ,{\mathcal {A}})\rightarrow ({\mathcal {X}},\Sigma )}"></span> are <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{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> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56204ada1ecf9dd5c6beddfdfd0f341cd69ff632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}"></span> is a sample from these 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 T_{n}:({\mathcal {X}}^{n},\Sigma ^{n})\rightarrow (\Gamma ,S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}:({\mathcal {X}}^{n},\Sigma ^{n})\rightarrow (\Gamma ,S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295393b1f53a9703233138a6fdde743bd1f98195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.822ex; height:2.843ex;" alt="{\displaystyle T_{n}:({\mathcal {X}}^{n},\Sigma ^{n})\rightarrow (\Gamma ,S)}"></span> is an estimator. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\dots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55cb337f87f66a675fe0c7c3d4ecaa801f989a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dots ,n\}}"></span>. The empirical influence function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EIF_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>I</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EIF_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/761e681ccffd81f516d1dc50a9a1b0a7e2d0de86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.242ex; height:2.509ex;" alt="{\displaystyle EIF_{i}}"></span> at observation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is defined by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EIF_{i}:x\in {\mathcal {X}}\mapsto n\cdot (T_{n}(x_{1},\dots ,x_{i-1},x,x_{i+1},\dots ,x_{n})-T_{n}(x_{1},\dots ,x_{i-1},x_{i},x_{i+1},\dots ,x_{n}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>I</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EIF_{i}:x\in {\mathcal {X}}\mapsto n\cdot (T_{n}(x_{1},\dots ,x_{i-1},x,x_{i+1},\dots ,x_{n})-T_{n}(x_{1},\dots ,x_{i-1},x_{i},x_{i+1},\dots ,x_{n}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ce3ea304a0a1d548d3f8a2648d3015b9713349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.424ex; height:2.843ex;" alt="{\displaystyle EIF_{i}:x\in {\mathcal {X}}\mapsto n\cdot (T_{n}(x_{1},\dots ,x_{i-1},x,x_{i+1},\dots ,x_{n})-T_{n}(x_{1},\dots ,x_{i-1},x_{i},x_{i+1},\dots ,x_{n}))}"></span></dd></dl> <p>What this means is that we are replacing the <i>i</i>-th value in the sample by an arbitrary value and looking at the output of the estimator. Alternatively, the EIF is defined as the effect, scaled by n+1 instead of n, on the estimator of adding the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> to the sample.<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. (July 2016)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Influence_function_and_sensitivity_curve">Influence function and sensitivity curve</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=13" title="Edit section: Influence function and sensitivity curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Biweight.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Biweight.svg/300px-Biweight.svg.png" decoding="async" width="300" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Biweight.svg/450px-Biweight.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Biweight.svg/600px-Biweight.svg.png 2x" data-file-width="649" data-file-height="589" /></a><figcaption>Influence function when Tukey's biweight function (see section M-estimators below) is used as a loss function. Points with large deviation have no influence (y=0).</figcaption></figure> <p>Instead of relying solely on the data, we could use the distribution of the random variables. The approach is quite different from that of the previous paragraph. What we are now trying to do is to see what happens to an estimator when we change the distribution of the data slightly: it assumes a <i>distribution,</i> and measures sensitivity to change in this distribution. By contrast, the empirical influence assumes a <i>sample set,</i> and measures sensitivity to change in the samples.<sup id="cite_ref-FOOTNOTEvon_Mises1947_9-0" class="reference"><a href="#cite_note-FOOTNOTEvon_Mises1947-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> be a convex subset of the set of all finite signed measures on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>. We want to estimate the parameter <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 \in \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \in \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc76a89f6f175a02e371f31f86a8906fa3fd5e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.739ex; height:2.176ex;" alt="{\displaystyle \theta \in \Theta }"></span> of a 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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> in <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Let the functional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:A\rightarrow \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:A\rightarrow \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d0dd40547760dffdc1b83dd6651bf9fdd918da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.383ex; height:2.176ex;" alt="{\displaystyle T:A\rightarrow \Gamma }"></span> be the asymptotic value of some estimator sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25fbf21de09aca7ee46e637c89b442ee86bd2fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.887ex; height:2.843ex;" alt="{\displaystyle (T_{n})_{n\in \mathbb {N} }}"></span>. We will suppose that this functional is <a href="/wiki/Fisher_consistency" title="Fisher consistency">Fisher consistent</a>, 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 \forall \theta \in \Theta ,T(F_{\theta })=\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \theta \in \Theta ,T(F_{\theta })=\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7d2b64083b03f1738ae03efad6d86ea2eacc67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.198ex; height:2.843ex;" alt="{\displaystyle \forall \theta \in \Theta ,T(F_{\theta })=\theta }"></span>. This means that at the model <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, the estimator sequence asymptotically measures the correct quantity. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> be some distribution in <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. What happens when the data doesn't follow the model <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> exactly but another, slightly different, "going towards" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>? </p><p>We're looking at: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dT_{G-F}(F)=\lim _{t\rightarrow 0^{+}}{\frac {T(tG+(1-t)F)-T(F)}{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mo>−</mo> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mi>G</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dT_{G-F}(F)=\lim _{t\rightarrow 0^{+}}{\frac {T(tG+(1-t)F)-T(F)}{t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c095967db1bea697b10bb5669eae1ad1a528fcf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.111ex; height:6.176ex;" alt="{\displaystyle dT_{G-F}(F)=\lim _{t\rightarrow 0^{+}}{\frac {T(tG+(1-t)F)-T(F)}{t}}}"></span>,</dd></dl> <p>which is the <a href="/wiki/One-sided_limit" title="One-sided limit">one-sided</a> <a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, in the direction of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G-F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>−</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G-F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f255c4d51e5e373bc3cf37a544bb4a956956e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle G-F}"></span>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathcal {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\mathcal {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3cd596d560b9f3bc71615911c2438cbbf49c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.046ex; height:2.176ex;" alt="{\displaystyle x\in {\mathcal {X}}}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bbf9f91a144687ca2a31307ad731850f8ee44aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.108ex; height:2.509ex;" alt="{\displaystyle \Delta _{x}}"></span> is the probability measure which gives mass 1 to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </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/a120eeb8a091b516595765bd08b306f2394e7721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}"></span>. We choose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\Delta _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\Delta _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e76b9f708da442cc2955807b7123612334fef6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.034ex; height:2.509ex;" alt="{\displaystyle G=\Delta _{x}}"></span>. The influence function is then defined by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IF(x;T;F):=\lim _{t\rightarrow 0^{+}}{\frac {T(t\Delta _{x}+(1-t)F)-T(F)}{t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IF(x;T;F):=\lim _{t\rightarrow 0^{+}}{\frac {T(t\Delta _{x}+(1-t)F)-T(F)}{t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50945e81064588433c9f5a4b8ee8787ea8de895c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.026ex; height:6.176ex;" alt="{\displaystyle IF(x;T;F):=\lim _{t\rightarrow 0^{+}}{\frac {T(t\Delta _{x}+(1-t)F)-T(F)}{t}}.}"></span></dd></dl> <p>It describes the effect of an infinitesimal contamination at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> on the estimate we are seeking, standardized by the mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> of the contamination (the asymptotic bias caused by contamination in the observations). For a robust estimator, we want a bounded influence function, that is, one which does not go to infinity as x becomes arbitrarily large. </p><p>The empirical influence function uses the <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22e0749dfc79fd15d8f156203a276fb7092fc51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.805ex; height:2.843ex;" alt="{\displaystyle {\hat {F}}}"></span> instead of the distribution function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, making use of the <a href="/w/index.php?title=Drop-in_principle&action=edit&redlink=1" class="new" title="Drop-in principle (page does not exist)">drop-in principle</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Desirable_properties">Desirable properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=14" title="Edit section: Desirable properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Properties of an influence function that bestow it with desirable performance are: </p> <ol><li>Finite rejection point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b9afc5b13d9b11ab7cc83f865daa1f7fbee033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.843ex;" alt="{\displaystyle \rho ^{*}}"></span>,</li> <li>Small gross-error sensitivity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94d9c54dd2c77e4571ed51bd635158b22c204a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}"></span>,</li> <li>Small local-shift sensitivity <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 \lambda ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfedb17605a2ff7c84fec07a334cb4d20171f9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.409ex; height:2.343ex;" alt="{\displaystyle \lambda ^{*}}"></span>.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Rejection_point">Rejection point</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=15" title="Edit section: Rejection point"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{*}:=\inf _{r>0}\{r:IF(x;T;F)=0,|x|>r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mo fence="false" stretchy="false">{</mo> <mi>r</mi> <mo>:</mo> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{*}:=\inf _{r>0}\{r:IF(x;T;F)=0,|x|>r\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8d349dca06776a864ed3b6fda04a561f7c7846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.716ex; height:4.009ex;" alt="{\displaystyle \rho ^{*}:=\inf _{r>0}\{r:IF(x;T;F)=0,|x|>r\}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Gross-error_sensitivity">Gross-error sensitivity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=16" title="Edit section: Gross-error sensitivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}(T;F):=\sup _{x\in {\mathcal {X}}}|IF(x;T;F)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}(T;F):=\sup _{x\in {\mathcal {X}}}|IF(x;T;F)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee674ad0f721a70020b74ca5634b174db5c08ac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.978ex; height:4.509ex;" alt="{\displaystyle \gamma ^{*}(T;F):=\sup _{x\in {\mathcal {X}}}|IF(x;T;F)|}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Local-shift_sensitivity">Local-shift sensitivity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=17" title="Edit section: Local-shift sensitivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{*}(T;F):=\sup _{(x,y)\in {\mathcal {X}}^{2} \atop x\neq y}\left\|{\frac {IF(y;T;F)-IF(x;T;F)}{y-x}}\right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </munder> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>;</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{*}(T;F):=\sup _{(x,y)\in {\mathcal {X}}^{2} \atop x\neq y}\left\|{\frac {IF(y;T;F)-IF(x;T;F)}{y-x}}\right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c140b9cd91a114e67752cf8d53844749268c406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:47.781ex; height:8.509ex;" alt="{\displaystyle \lambda ^{*}(T;F):=\sup _{(x,y)\in {\mathcal {X}}^{2} \atop x\neq y}\left\|{\frac {IF(y;T;F)-IF(x;T;F)}{y-x}}\right\|}"></span></dd></dl> <p>This value, which looks a lot like a <a href="/wiki/Lipschitz_constant" class="mw-redirect" title="Lipschitz constant">Lipschitz constant</a>, represents the effect of shifting an observation slightly from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <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> to a neighbouring point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, i.e., add an observation at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and remove one at <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>. </p> <div class="mw-heading mw-heading2"><h2 id="M-estimators">M-estimators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=18" title="Edit section: M-estimators"><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/M-estimator" title="M-estimator">M-estimator</a></div> <p><i>(The mathematical context of this paragraph is given in the section on empirical influence functions.)</i> </p><p>Historically, several approaches to robust estimation were proposed, including R-estimators and <a href="/wiki/L-estimator" title="L-estimator">L-estimators</a>. However, M-estimators now appear to dominate the field as a result of their generality, their potential for high breakdown points and comparatively high efficiency. See <a href="#CITEREFHuber1981">Huber (1981)</a>. </p><p>M-estimators are not inherently robust. However, they can be designed to achieve favourable properties, including robustness. M-estimator are a generalization of <a href="/wiki/Maximum_likelihood_estimator" class="mw-redirect" title="Maximum likelihood estimator">maximum likelihood estimators</a> (MLEs) which is determined by maximizing <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="{\textstyle \prod _{i=1}^{n}f(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{i=1}^{n}f(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e801a5de66232c2bb4b7721524e366be173a4faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.698ex; height:3.176ex;" alt="{\textstyle \prod _{i=1}^{n}f(x_{i})}"></span> or, equivalently, minimizing <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="{\textstyle \sum _{i=1}^{n}-\log f(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}-\log f(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1940fde939c40e3b8ebe1c68764cd42c0f386d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.512ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}-\log f(x_{i})}"></span>. In 1964, Huber proposed to generalize this to the minimization of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{n}\rho (x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}\rho (x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637f21d2f3919de16868b96069b7dacd973fde07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.882ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}\rho (x_{i})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is some function. MLE are therefore a special case of M-estimators (hence the name: "<i>M</i>aximum likelihood type" estimators). </p><p>Minimizing <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="{\textstyle \sum _{i=1}^{n}\rho (x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}\rho (x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637f21d2f3919de16868b96069b7dacd973fde07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.882ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}\rho (x_{i})}"></span> can often be done by differentiating <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> and solving <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="{\textstyle \sum _{i=1}^{n}\psi (x_{i})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}\psi (x_{i})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6171db7b5236a96fc938158908cfc05954bd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.454ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}\psi (x_{i})=0}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi (x)={\frac {d\rho (x)}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi (x)={\frac {d\rho (x)}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67cfd879fb1433118703d8f9e097a61f27ddf86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.516ex; height:4.343ex;" alt="{\textstyle \psi (x)={\frac {d\rho (x)}{dx}}}"></span> (if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> has a derivative). </p><p>Several choices of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> have been proposed. The two figures below show four <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> functions and their corresponding <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> functions. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:RhoFunctions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/RhoFunctions.png" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="400" data-file-height="300" /></a></span> </p><p>For squared errors, <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 \rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32a26d8795457b2f5c2bdc078758dcbbc71b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \rho (x)}"></span> increases at an accelerating rate, whilst for absolute errors, it increases at a constant rate. When Winsorizing is used, a mixture of these two effects is introduced: for small values of x, <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> increases at the squared rate, but once the chosen threshold is reached (1.5 in this example), the rate of increase becomes constant. This Winsorised estimator is also known as the <a href="/wiki/Huber_loss_function" class="mw-redirect" title="Huber loss function">Huber loss function</a>. </p><p>Tukey's biweight (also known as bisquare) function behaves in a similar way to the squared error function at first, but for larger errors, the function tapers off. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:PsiFunctions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/78/PsiFunctions.png" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="400" data-file-height="300" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Properties_of_M-estimators">Properties of M-estimators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=19" title="Edit section: Properties of M-estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>M-estimators do not necessarily relate to a probability density function. Therefore, off-the-shelf approaches to inference that arise from likelihood theory can not, in general, be used. </p><p>It can be shown that M-estimators are asymptotically normally distributed so that as long as their standard errors can be computed, an approximate approach to inference is available. </p><p>Since M-estimators are normal only asymptotically, for small sample sizes it might be appropriate to use an alternative approach to inference, such as the bootstrap. However, M-estimates are not necessarily unique (i.e., there might be more than one solution that satisfies the equations). Also, it is possible that any particular bootstrap sample can contain more outliers than the estimator's breakdown point. Therefore, some care is needed when designing bootstrap schemes. </p><p>Of course, as we saw with the speed-of-light example, the mean is only normally distributed asymptotically and when outliers are present the approximation can be very poor even for quite large samples. However, classical statistical tests, including those based on the mean, are typically bounded above by the nominal size of the test. The same is not true of M-estimators and the type I error rate can be substantially above the nominal level. </p><p>These considerations do not "invalidate" M-estimation in any way. They merely make clear that some care is needed in their use, as is true of any other method of estimation. </p> <div class="mw-heading mw-heading3"><h3 id="Influence_function_of_an_M-estimator">Influence function of an M-estimator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=20" title="Edit section: Influence function of an M-estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It can be shown that the influence function of an M-estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> which means we can derive the properties of such an estimator (such as its rejection point, gross-error sensitivity or local-shift sensitivity) when we know its <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> function. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IF(x;T,F)=M^{-1}\psi (x,T(F))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IF(x;T,F)=M^{-1}\psi (x,T(F))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c347eecdf33d27507f473802090003506f7706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.299ex; height:3.176ex;" alt="{\displaystyle IF(x;T,F)=M^{-1}\psi (x,T(F))}"></span></dd></dl> <p>with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"></span> given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=-\int _{\mathcal {X}}\left({\frac {\partial \psi (x,\theta )}{\partial \theta }}\right)_{T(F)}\,dF(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">X</mi> </mrow> </mrow> </msub> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=-\int _{\mathcal {X}}\left({\frac {\partial \psi (x,\theta )}{\partial \theta }}\right)_{T(F)}\,dF(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4dd0be9fdc476ac1e0830af4f47903b0ea1f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.355ex; height:6.843ex;" alt="{\displaystyle M=-\int _{\mathcal {X}}\left({\frac {\partial \psi (x,\theta )}{\partial \theta }}\right)_{T(F)}\,dF(x).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Choice_of_ψ_and_ρ"><span id="Choice_of_.CF.88_and_.CF.81"></span>Choice of <i>ψ</i> and <i>ρ</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=21" title="Edit section: Choice of ψ and ρ"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many practical situations, the choice of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> function is not critical to gaining a good robust estimate, and many choices will give similar results that offer great improvements, in terms of efficiency and bias, over classical estimates in the presence of outliers.<sup id="cite_ref-FOOTNOTEHuber1981_11-0" class="reference"><a href="#cite_note-FOOTNOTEHuber1981-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Theoretically, <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> functions are to be preferred,<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="are some psi functions preferred over other psi functions, and if so, which? Or are psi functions preferred over something else, and if so, what? (October 2019)">clarification needed</span></a></i>]</sup> and Tukey's biweight (also known as bisquare) function is a popular choice.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> recommend the biweight function with efficiency at the normal set to 85%. </p> <div class="mw-heading mw-heading2"><h2 id="Robust_parametric_approaches">Robust parametric approaches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=22" title="Edit section: Robust parametric approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>M-estimators do not necessarily relate to a density function and so are not fully parametric. Fully parametric approaches to robust modeling and inference, both Bayesian and likelihood approaches, usually deal with heavy-tailed distributions such as Student's <i>t</i>-distribution. </p><p>For the <i>t</i>-distribution with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> degrees of freedom, it can be shown that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)={\frac {x}{x^{2}+\nu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>ν</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)={\frac {x}{x^{2}+\nu }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a931d49c499e1202192b83b0b733086962ec16ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.69ex; height:5.176ex;" alt="{\displaystyle \psi (x)={\frac {x}{x^{2}+\nu }}.}"></span></dd></dl> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ca7fd51cd02b2c4311446c6d6b0f53b2c529cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =1}"></span>, the <i>t</i>-distribution is equivalent to the Cauchy distribution. The degrees of freedom is sometimes known as the <i>kurtosis parameter</i>. It is the parameter that controls how heavy the tails are. In principle, <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> can be estimated from the data in the same way as any other parameter. In practice, it is common for there to be multiple local maxima when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is allowed to vary. As such, it is common to fix <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> at a value around 4 or 6. The figure below displays the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>-function for 4 different values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:TDistPsi.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/TDistPsi.png" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="400" data-file-height="300" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Example:_speed-of-light_data_2">Example: speed-of-light data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=23" title="Edit section: Example: speed-of-light data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the speed-of-light data, allowing the kurtosis parameter to vary and maximizing the likelihood, we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}=27.40,\quad {\hat {\sigma }}=3.81,\quad {\hat {\nu }}=2.13.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>27.40</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>3.81</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2.13.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}=27.40,\quad {\hat {\sigma }}=3.81,\quad {\hat {\nu }}=2.13.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1864c3e4aa880bdf14dd77634345da15ed03219c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.268ex; height:2.676ex;" alt="{\displaystyle {\hat {\mu }}=27.40,\quad {\hat {\sigma }}=3.81,\quad {\hat {\nu }}=2.13.}"></span></dd></dl> <p>Fixing <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 \nu =4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb9572c6b72be8bf95cde8355eb9860168e6cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =4}"></span> and maximizing the likelihood gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}=27.49,\quad {\hat {\sigma }}=4.51.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>27.49</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>4.51.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}=27.49,\quad {\hat {\sigma }}=4.51.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e59db1b95d2c30ff86f9a6c3cd4dd139e769f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.363ex; height:2.676ex;" alt="{\displaystyle {\hat {\mu }}=27.49,\quad {\hat {\sigma }}=4.51.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=24" title="Edit section: Related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">pivotal quantity</a> is a function of data, whose underlying population distribution is a member of a parametric family, that is not dependent on the values of the parameters. An <a href="/wiki/Ancillary_statistic" title="Ancillary statistic">ancillary statistic</a> is such a function that is also a statistic, meaning that it is computed in terms of the data alone. Such functions are robust to parameters in the sense that they are independent of the values of the parameters, but not robust to the model in the sense that they assume an underlying model (parametric family), and in fact, such functions are often very sensitive to violations of the model assumptions. Thus <a href="/wiki/Test_statistic" title="Test statistic">test statistics</a>, frequently constructed in terms of these to not be sensitive to assumptions about parameters, are still very sensitive to model assumptions. </p> <div class="mw-heading mw-heading2"><h2 id="Replacing_outliers_and_missing_values">Replacing outliers and missing values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=25" title="Edit section: Replacing outliers and missing values"><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/Imputation_(statistics)" title="Imputation (statistics)">Imputation (statistics)</a></div> <p>Replacing <a href="/wiki/Missing_data" title="Missing data">missing data</a> is called <a href="/wiki/Imputation_(statistics)" title="Imputation (statistics)">imputation</a>. If there are relatively few missing points, there are some models which can be used to estimate values to complete the series, such as replacing missing values with the mean or median of the data. <a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a> can also be used to estimate missing values.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In addition, <a href="/wiki/Outliers" class="mw-redirect" title="Outliers">outliers</a> can sometimes be accommodated in the data through the use of trimmed means, other scale estimators apart from standard deviation (e.g., MAD) and Winsorization.<sup id="cite_ref-FOOTNOTEMcBeanRovers1998_14-0" class="reference"><a href="#cite_note-FOOTNOTEMcBeanRovers1998-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In calculations of a trimmed mean, a fixed percentage of data is dropped from each end of an ordered data, thus eliminating the outliers. The mean is then calculated using the remaining data. <a href="/wiki/Winsorizing" title="Winsorizing">Winsorizing</a> involves accommodating an outlier by replacing it with the next highest or next smallest value as appropriate.<sup id="cite_ref-FOOTNOTERustumAdeloye2007_15-0" class="reference"><a href="#cite_note-FOOTNOTERustumAdeloye2007-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>However, using these types of models to predict missing values or outliers in a long time series is difficult and often unreliable, particularly if the number of values to be in-filled is relatively high in comparison with total record length. The accuracy of the estimate depends on how good and representative the model is and how long the period of missing values extends.<sup id="cite_ref-FOOTNOTERosenLennox2001_16-0" class="reference"><a href="#cite_note-FOOTNOTERosenLennox2001-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> When dynamic evolution is assumed in a series, the missing data point problem becomes an exercise in <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">multivariate analysis</a> (rather than the univariate approach of most traditional methods of estimating missing values and outliers). In such cases, a multivariate model will be more representative than a univariate one for predicting missing values. The <a href="/wiki/Self-organizing_map" title="Self-organizing map">Kohonen self organising map</a> (KSOM) offers a simple and robust multivariate model for data analysis, thus providing good possibilities to estimate missing values, taking into account their relationship or correlation with other pertinent variables in the data record.<sup id="cite_ref-FOOTNOTERustumAdeloye2007_15-1" class="reference"><a href="#cite_note-FOOTNOTERustumAdeloye2007-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Standard <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filters</a> are not robust to outliers. To this end <a href="#CITEREFTingTheodorouSchaal2007">Ting, Theodorou & Schaal (2007)</a> have recently shown that a modification of <a href="/wiki/Masreliez%27s_theorem" title="Masreliez's theorem">Masreliez's theorem</a> can deal with outliers. </p><p>One common approach to handle outliers in data analysis is to perform outlier detection first, followed by an efficient estimation method (e.g., the least squares). While this approach is often useful, one must keep in mind two challenges. First, an outlier detection method that relies on a non-robust initial fit can suffer from the effect of masking, that is, a group of outliers can mask each other and escape detection.<sup id="cite_ref-FOOTNOTERousseeuwLeroy1987_17-0" class="reference"><a href="#cite_note-FOOTNOTERousseeuwLeroy1987-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Second, if a high breakdown initial fit is used for outlier detection, the follow-up analysis might inherit some of the inefficiencies of the initial estimator.<sup id="cite_ref-FOOTNOTEHePortnoy1992_18-0" class="reference"><a href="#cite_note-FOOTNOTEHePortnoy1992-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Use_in_machine_learning">Use in machine learning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_statistics&action=edit&section=26" title="Edit section: Use in machine learning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although influence functions have a long history in statistics, they were not widely used in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> due to several challenges. One of the primary obstacles is that traditional influence functions rely on expensive second-order derivative computations and assume model differentiability and convexity. These assumptions are limiting, especially in modern machine learning, where models are often non-differentiable, non-convex, and operate in high-dimensional spaces. </p><p><a href="#CITEREFKohLiang2017">Koh & Liang (2017)</a> addressed these challenges by introducing methods to efficiently approximate influence functions using second-order optimization techniques, such as those developed by <a href="#CITEREFPearlmutter1994">Pearlmutter (1994)</a>, <a href="#CITEREFMartens2010">Martens (2010)</a>, and <a href="#CITEREFAgarwalBullinsHazan2017">Agarwal, Bullins & Hazan (2017)</a>. Their approach remains effective even when the assumptions of differentiability and convexity degrade, enabling influence functions to be used in the context of non-convex deep learning models. They demonstrated that influence functions are a powerful and versatile tool that can be applied to a variety of tasks in machine learning, including: </p> <ul><li>Understanding Model Behavior: Influence functions help identify which training points are most “responsible” for a given prediction, offering insights into how models generalize from training data.</li></ul> <ul><li>Debugging Models: Influence functions can assist in identifying domain mismatches—when the training data distribution does not match the test data distribution—which can cause models with high training accuracy to perform poorly on test data, as shown by <a href="#CITEREFBen-DavidBlitzerCrammerKulesza2010">Ben-David et al. (2010)</a>. By revealing which training examples contribute most to errors, developers can address these mismatches.</li></ul> <ul><li>Dataset Error Detection: Noisy or corrupted labels are common in real-world data, especially when crowdsourced or adversarially attacked. Influence functions allow human experts to prioritize reviewing only the most impactful examples in the training set, facilitating efficient error detection and correction.</li></ul> <ul><li>Adversarial Attacks: Models that rely heavily on a small number of influential training points are vulnerable to adversarial perturbations. These perturbed inputs can significantly alter predictions and pose security risks in machine learning systems where attackers have access to the training data (See <a href="/wiki/Adversarial_machine_learning" title="Adversarial machine learning">adversarial machine learning</a>).</li></ul> <p>Koh and Liang’s contributions have opened the door for influence functions to be used in various applications across machine learning, from interpretability to security, marking a significant advance in their applicability. </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=Robust_statistics&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Robust_confidence_intervals" class="mw-redirect" title="Robust confidence intervals">Robust confidence intervals</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust regression</a></li> <li><a href="/wiki/Unit-weighted_regression" title="Unit-weighted regression">Unit-weighted regression</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=Robust_statistics&action=edit&section=28" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSarkar2014" class="citation journal cs1">Sarkar, Palash (2014-05-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0378375813001110">"On some connections between statistics and cryptology"</a>. <i>Journal of Statistical Planning and Inference</i>. <b>148</b>: 20–37. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jspi.2013.05.008">10.1016/j.jspi.2013.05.008</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/0378-3758">0378-3758</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+Statistical+Planning+and+Inference&rft.atitle=On+some+connections+between+statistics+and+cryptology&rft.volume=148&rft.pages=20-37&rft.date=2014-05-01&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2013.05.008&rft.issn=0378-3758&rft.aulast=Sarkar&rft.aufirst=Palash&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0378375813001110&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></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 id="CITEREFHuberRonchetti2009" class="citation book cs1">Huber, Peter J.; Ronchetti, Elvezio M. (2009-01-29). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9780470434697"><i>Robust Statistics</i></a>. Wiley Series in Probability and Statistics (1 ed.). Wiley. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9780470434697">10.1002/9780470434697</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-12990-6" title="Special:BookSources/978-0-470-12990-6"><bdi>978-0-470-12990-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+Statistics&rft.series=Wiley+Series+in+Probability+and+Statistics&rft.edition=1&rft.pub=Wiley&rft.date=2009-01-29&rft_id=info%3Adoi%2F10.1002%2F9780470434697&rft.isbn=978-0-470-12990-6&rft.aulast=Huber&rft.aufirst=Peter+J.&rft.au=Ronchetti%2C+Elvezio+M.&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fbook%2F10.1002%2F9780470434697&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" 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="CITEREFHuberRonchetti2009" class="citation book cs1">Huber, Peter J.; Ronchetti, Elvezio M. (2009-01-29). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9780470434697"><i>Robust Statistics</i></a>. Wiley Series in Probability and Statistics (1 ed.). Wiley. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9780470434697">10.1002/9780470434697</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-12990-6" title="Special:BookSources/978-0-470-12990-6"><bdi>978-0-470-12990-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+Statistics&rft.series=Wiley+Series+in+Probability+and+Statistics&rft.edition=1&rft.pub=Wiley&rft.date=2009-01-29&rft_id=info%3Adoi%2F10.1002%2F9780470434697&rft.isbn=978-0-470-12990-6&rft.aulast=Huber&rft.aufirst=Peter+J.&rft.au=Ronchetti%2C+Elvezio+M.&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fbook%2F10.1002%2F9780470434697&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></span> </li> <li id="cite_note-huber-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-huber_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-huber_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-huber_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHuber1981">Huber (1981)</a>, page 1.</span> </li> <li id="cite_note-FOOTNOTERousseeuwCroux1993-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERousseeuwCroux1993_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRousseeuwCroux1993">Rousseeuw & Croux (1993)</a>.</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 id="CITEREFMasters" class="citation web cs1">Masters, Jeffrey. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160915081133/https://www.wunderground.com/climate/holefaq.asp">"When was the ozone hole discovered"</a>. <i>Weather Underground</i>. Archived from <a rel="nofollow" class="external text" href="http://www.wunderground.com/climate/holefaq.asp">the original</a> on 2016-09-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Weather+Underground&rft.atitle=When+was+the+ozone+hole+discovered&rft.aulast=Masters&rft.aufirst=Jeffrey&rft_id=http%3A%2F%2Fwww.wunderground.com%2Fclimate%2Fholefaq.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></span> </li> <li id="cite_note-maronna-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-maronna_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaronnaMartinYohaiSalibián-Barrera2019">Maronna et al. (2019)</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://secamlocal.ex.ac.uk/people/staff/dbs202/cag/courses/MT37C/course/node13.html">Resistant statistics</a>, <a rel="nofollow" class="external text" href="http://secamlocal.ex.ac.uk/people/staff/dbs202/">David B. Stephenson</a></span> </li> <li id="cite_note-FOOTNOTEvon_Mises1947-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvon_Mises1947_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvon_Mises1947">von Mises (1947)</a>.</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"><a href="#CITEREFHuber1981">Huber (1981)</a>, page 45</span> </li> <li id="cite_note-FOOTNOTEHuber1981-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHuber1981_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHuber1981">Huber (1981)</a>.</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="#CITEREFMaronnaMartinYohaiSalibián-Barrera2019">Maronna et al. (2019)</a></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"><a href="#CITEREFMacDonaldZucchini1997">MacDonald & Zucchini (1997)</a>; <a href="#CITEREFHarveyFernandes1989">Harvey & Fernandes (1989)</a>.</span> </li> <li id="cite_note-FOOTNOTEMcBeanRovers1998-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMcBeanRovers1998_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcBeanRovers1998">McBean & Rovers (1998)</a>.</span> </li> <li id="cite_note-FOOTNOTERustumAdeloye2007-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERustumAdeloye2007_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERustumAdeloye2007_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRustumAdeloye2007">Rustum & Adeloye (2007)</a>.</span> </li> <li id="cite_note-FOOTNOTERosenLennox2001-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERosenLennox2001_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRosenLennox2001">Rosen & Lennox (2001)</a>.</span> </li> <li id="cite_note-FOOTNOTERousseeuwLeroy1987-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERousseeuwLeroy1987_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRousseeuwLeroy1987">Rousseeuw & Leroy (1987)</a>.</span> </li> <li id="cite_note-FOOTNOTEHePortnoy1992-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHePortnoy1992_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHePortnoy1992">He & Portnoy (1992)</a>.</span> </li> </ol></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=Robust_statistics&action=edit&section=29" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarcomeniGreco2013" class="citation cs2">Farcomeni, A.; Greco, L. (2013), <i>Robust methods for data reduction</i>, Boca Raton, FL: Chapman & Hall/CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-9062-5" title="Special:BookSources/978-1-4665-9062-5"><bdi>978-1-4665-9062-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+methods+for+data+reduction&rft.place=Boca+Raton%2C+FL&rft.pub=Chapman+%26+Hall%2FCRC+Press&rft.date=2013&rft.isbn=978-1-4665-9062-5&rft.aulast=Farcomeni&rft.aufirst=A.&rft.au=Greco%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHampelRonchettiRousseeuwStahel1986" class="citation cs2"><a href="/wiki/Frank_Hampel" title="Frank Hampel">Hampel, Frank R.</a>; Ronchetti, Elvezio M.; <a href="/wiki/Peter_J._Rousseeuw" class="mw-redirect" title="Peter J. Rousseeuw">Rousseeuw, Peter J.</a>; Stahel, Werner A. (1986), <i>Robust statistics</i>, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, New York: John Wiley & Sons, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-82921-8" title="Special:BookSources/0-471-82921-8"><bdi>0-471-82921-8</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=0829458">0829458</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+statistics&rft.place=New+York&rft.series=Wiley+Series+in+Probability+and+Mathematical+Statistics%3A+Probability+and+Mathematical+Statistics&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1986&rft.isbn=0-471-82921-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D829458%23id-name%3DMR&rft.aulast=Hampel&rft.aufirst=Frank+R.&rft.au=Ronchetti%2C+Elvezio+M.&rft.au=Rousseeuw%2C+Peter+J.&rft.au=Stahel%2C+Werner+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span>. Republished in paperback, 2005.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarveyFernandes1989" class="citation cs2">Harvey, A. C.; Fernandes, C. 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(1998), <i>Robust nonparametric statistical methods</i>, Kendall's Library of Statistics, vol. 5, New York: John Wiley & Sons, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-340-54937-8" title="Special:BookSources/0-340-54937-8"><bdi>0-340-54937-8</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=1604954">1604954</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+nonparametric+statistical+methods&rft.place=New+York&rft.series=Kendall%27s+Library+of+Statistics&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1998&rft.isbn=0-340-54937-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1604954%23id-name%3DMR&rft.aulast=Hettmansperger&rft.aufirst=T.+P.&rft.au=McKean%2C+J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span>. 2nd ed., CRC Press, 2011.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuber1981" class="citation cs2"><a href="/wiki/Peter_J._Huber" title="Peter J. 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International Conference on Machine Learning. Haifa, Israel: PMLR. pp. 735–742. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781605589077" title="Special:BookSources/9781605589077"><bdi>9781605589077</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Deep+learning+via+Hessian-free+optimization&rft.place=Haifa%2C+Israel&rft.pages=735-742&rft.pub=PMLR&rft.date=2010&rft.isbn=9781605589077&rft.aulast=Martens&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgarwalBullinsHazan2017" class="citation cs2">Agarwal, Naman; Bullins, Brian; Hazan, Elad (2017), "Second-order stochastic optimization for machine learning in linear time", <i>Journal of Machine Learning Research</i>, <b>18</b> (116): 1–40, <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/1602.03943">1602.03943</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Machine+Learning+Research&rft.atitle=Second-order+stochastic+optimization+for+machine+learning+in+linear+time&rft.volume=18&rft.issue=116&rft.pages=1-40&rft.date=2017&rft_id=info%3Aarxiv%2F1602.03943&rft.aulast=Agarwal&rft.aufirst=Naman&rft.au=Bullins%2C+Brian&rft.au=Hazan%2C+Elad&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen-DavidBlitzerCrammerKulesza2010" class="citation cs2">Ben-David, Shai; Blitzer, John; Crammer, Koby; Kulesza, Alex; Pereira, Fernando; Vaughan, Jennifer Wortman (2010), "A theory of learning from different domains", <i>Machine Learning</i>, <b>79</b> (1): 151–175, <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/10.1007%2Fs10994-009-5152-4">10.1007/s10994-009-5152-4</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1573-0565">1573-0565</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Machine+Learning&rft.atitle=A+theory+of+learning+from+different+domains&rft.volume=79&rft.issue=1&rft.pages=151-175&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2Fs10994-009-5152-4&rft.issn=1573-0565&rft.aulast=Ben-David&rft.aufirst=Shai&rft.au=Blitzer%2C+John&rft.au=Crammer%2C+Koby&rft.au=Kulesza%2C+Alex&rft.au=Pereira%2C+Fernando&rft.au=Vaughan%2C+Jennifer+Wortman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+statistics" class="Z3988"></span></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=Robust_statistics&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Brian_D._Ripley" title="Brian D. Ripley">Brian Ripley's</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121021081319/http://www.stats.ox.ac.uk/pub/StatMeth/Robust.pdf">robust statistics course notes.</a></li> <li><a rel="nofollow" class="external text" href="http://www.nickfieller.staff.shef.ac.uk/sheff-only/StatModall05.pdf">Nick Fieller's course notes on Statistical Modelling and Computation</a> contain material on robust regression.</li> <li><a rel="nofollow" class="external text" href="http://lagrange.math.siu.edu/Olive/ol-bookp.htm">David Olive's site</a> contains course notes on robust statistics and some data sets.</li> <li><a rel="nofollow" class="external text" href="http://jsxgraph.uni-bayreuth.de/wiki/index.php/Analyze_data_with_the_Statistics_software_R">Online experiments using R and JSXGraph</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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.navbar{display:none!important}}</style><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 href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a 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 class="mw-selflink selflink">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 trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li> <li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Environmental_statistics" 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