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approaches</span> </div> </a> <button aria-controls="toc-Outline_of_different_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 Outline of different approaches subsection</span> </button> <ul id="toc-Outline_of_different_approaches-sublist" class="vector-toc-list"> <li id="toc-Behrens_and_Fisher_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Behrens_and_Fisher_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Behrens and Fisher approach</span> </div> </a> <ul id="toc-Behrens_and_Fisher_approach-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Welch's_approximate_t_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Welch's_approximate_t_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Welch's approximate t solution</span> </div> </a> <ul id="toc-Welch's_approximate_t_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_approaches" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_approaches"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Other approaches</span> </div> </a> <ul id="toc-Other_approaches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exact_solutions_to_the_common_and_generalized_Behrens–Fisher_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exact_solutions_to_the_common_and_generalized_Behrens–Fisher_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Exact solutions to the common and generalized Behrens–Fisher problems</span> </div> </a> <ul id="toc-Exact_solutions_to_the_common_and_generalized_Behrens–Fisher_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Variants" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variants"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Variants</span> </div> </a> <ul id="toc-Variants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalisations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalisations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generalisations</span> </div> </a> <ul id="toc-Generalisations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical problem</div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-statistics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-statistics">Unsolved problem in statistics</span>:</div> <div class="unsolved-body">Is an approximation analogous to Fisher's argument necessary to solve the Behrens–Fisher problem?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_statistics" title="List of unsolved problems in statistics">(more unsolved problems in statistics)</a></div> </div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>Behrens–Fisher problem</b>, named after <a href="/wiki/Walter-Ulrich_Behrens" title="Walter-Ulrich Behrens">Walter-Ulrich Behrens</a> and <a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a>, is the problem of <a href="/wiki/Interval_estimation" title="Interval estimation">interval estimation</a> and <a href="/wiki/Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a> concerning the difference between the means of two <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a> populations when the <a href="/wiki/Variance" title="Variance">variances</a> of the two populations are not assumed to be equal, based on two <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> samples. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Specification">Specification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=1" title="Edit section: Specification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One difficulty with discussing the Behrens–Fisher problem and proposed solutions, is that there are many different interpretations of what is meant by "the Behrens–Fisher problem". These differences involve not only what is counted as being a relevant solution, but even the basic statement of the context being considered. </p> <div class="mw-heading mw-heading3"><h3 id="Context">Context</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=2" title="Edit section: Context"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub> and <i>Y</i><sub>1</sub>, ..., <i>Y</i><sub><i>m</i></sub> be <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> samples from two populations which both come from the same <a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">location–scale family</a> of distributions. The scale parameters are assumed to be unknown and not necessarily equal, and the problem is to assess whether the location parameters can reasonably be treated as equal. Lehmann<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> states that "the Behrens–Fisher problem" is used both for this general form of model when the family of distributions is arbitrary, and for when the restriction to a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> is made. While Lehmann discusses a number of approaches to the more general problem, mainly based on nonparametrics,<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> most other sources appear to use "the Behrens–Fisher problem" to refer only to the case where the distribution is assumed to be normal: most of this article makes this assumption. </p> <div class="mw-heading mw-heading3"><h3 id="Requirements_of_solutions">Requirements of solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=3" title="Edit section: Requirements of solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Solutions to the Behrens–Fisher problem have been presented that make use of either a <a href="/wiki/Frequentist_inference" title="Frequentist inference">classical</a> or a <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> point of view and either solution would be notionally invalid judged from the other point of view. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. Where exactness of the significance levels of statistical tests is required, there may be an additional requirement that the procedure should make maximum use of the statistical information in the dataset. It is well known that an exact test can be gained by randomly discarding data from the larger dataset until the sample sizes are equal, assembling data in pairs and taking differences, and then using an ordinary <a href="/wiki/T-test" class="mw-redirect" title="T-test">t-test</a> to test for the mean-difference being zero: clearly this would not be "optimal" in any sense. </p><p>The task of specifying interval estimates for this problem is one where a frequentist approach fails to provide an exact solution, although some approximations are available. Standard Bayesian approaches also fail to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found.<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 2014)">citation needed</span></a></i>]</sup> Thus study of the problem can be used to elucidate the differences between the frequentist and Bayesian approaches to interval estimation. </p> <div class="mw-heading mw-heading2"><h2 id="Outline_of_different_approaches">Outline of different approaches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=4" title="Edit section: Outline of different approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Behrens_and_Fisher_approach">Behrens and Fisher approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=5" title="Edit section: Behrens and Fisher approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a> in 1935 introduced <a href="/wiki/Fiducial_inference" title="Fiducial inference">fiducial inference</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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> in order to apply it to this problem. He referred to an earlier paper by <a href="/wiki/Walter-Ulrich_Behrens" title="Walter-Ulrich Behrens">Walter-Ulrich Behrens</a> from 1929. Behrens and Fisher proposed to find the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of </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 T\equiv {{\bar {x}}_{1}-{\bar {x}}_{2} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\equiv {{\bar {x}}_{1}-{\bar {x}}_{2} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d20318cf1f4f838600ef63756d653ddd03c149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.247ex; height:7.843ex;" alt="{\displaystyle T\equiv {{\bar {x}}_{1}-{\bar {x}}_{2} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6b2c89bbe61dcd9b39f1914d7ad9273b7b5be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle {\bar {x}}_{1}}"></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 {\bar {x}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc7c71cdf5ac5e985d8fd8cafe28226ced7bfc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle {\bar {x}}_{2}}"></span> are the two <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample means</a>, and <i>s</i><sub>1</sub> and <i>s</i><sub>2</sub> are their <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a>. See <a href="/wiki/Behrens%E2%80%93Fisher_distribution" title="Behrens–Fisher distribution">Behrens–Fisher distribution</a>. Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations, </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 {s_{1}/{\sqrt {n_{1}}} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {s_{1}/{\sqrt {n_{1}}} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f0c03e97ad862ac86dd6fe7bc2494788af8807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:18.159ex; height:8.676ex;" alt="{\displaystyle {s_{1}/{\sqrt {n_{1}}} \over {\sqrt {s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}}}.}"></span></dd></dl> <p>Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be <a href="/wiki/Significance_level" class="mw-redirect" title="Significance level">rejected with probability α</a> if the means were in fact equal. Many other methods of treating the problem have been proposed since, and the effect on the resulting confidence intervals have been investigated.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Welch's_approximate_t_solution"><span id="Welch.27s_approximate_t_solution"></span>Welch's approximate t solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=6" title="Edit section: Welch's approximate t solution"><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 articles: <a href="/wiki/Welch%27s_t_test" class="mw-redirect" title="Welch's t test">Welch's t test</a> and <a href="/wiki/Welch%E2%80%93Satterthwaite_equation" title="Welch–Satterthwaite equation">Welch–Satterthwaite equation</a></div> <p>A widely used method is that of <a href="/wiki/Bernard_Lewis_Welch" title="Bernard Lewis Welch">B. L. Welch</a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> who, like Fisher, was at <a href="/wiki/University_College_London" title="University College London">University College London</a>. The variance of the mean difference </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {d}}={\bar {x}}_{1}-{\bar {x}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {d}}={\bar {x}}_{1}-{\bar {x}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1989a890d3228d6ef2d158a8c8ac6cf8bdca3ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.288ex; height:2.843ex;" alt="{\displaystyle {\bar {d}}={\bar {x}}_{1}-{\bar {x}}_{2}}"></span></dd></dl> <p>results in </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 s_{\bar {d}}^{2}={\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\bar {d}}^{2}={\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3d89dbadb1eb55840425e052e39382e2a62cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.711ex; height:6.176ex;" alt="{\displaystyle s_{\bar {d}}^{2}={\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}.}"></span></dd></dl> <p>Welch (1938) approximated the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\bar {d}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\bar {d}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763430a476e805767ccdff65d866604514e063f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.555ex; height:3.509ex;" alt="{\displaystyle s_{\bar {d}}^{2}}"></span> by the Type III <a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson distribution</a> (a scaled <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a>) whose first two <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> agree with that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\bar {d}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\bar {d}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763430a476e805767ccdff65d866604514e063f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.555ex; height:3.509ex;" alt="{\displaystyle s_{\bar {d}}^{2}}"></span>. This applies to the following number of degrees of freedom (d.f.), which is generally non-integer: </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 \nu \approx {(\gamma _{1}+\gamma _{2})^{2} \over \gamma _{1}^{2}/(n_{1}-1)+\gamma _{2}^{2}/(n_{2}-1)}\quad {\text{ where }}\gamma _{i}=\sigma _{i}^{2}/n_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \approx {(\gamma _{1}+\gamma _{2})^{2} \over \gamma _{1}^{2}/(n_{1}-1)+\gamma _{2}^{2}/(n_{2}-1)}\quad {\text{ where }}\gamma _{i}=\sigma _{i}^{2}/n_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c58f6cf53599c1328c9e5a65ea97a21a91ece15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:52.444ex; height:6.843ex;" alt="{\displaystyle \nu \approx {(\gamma _{1}+\gamma _{2})^{2} \over \gamma _{1}^{2}/(n_{1}-1)+\gamma _{2}^{2}/(n_{2}-1)}\quad {\text{ where }}\gamma _{i}=\sigma _{i}^{2}/n_{i}.}"></span></dd></dl> <p>Under the null hypothesis of equal expectations, <span class="nowrap"><i>μ</i><sub>1</sub> = <i>μ</i><sub>2</sub></span>, the distribution of the Behrens–Fisher statistic <i>T</i>, which also depends on the variance ratio <i>σ</i><sub>1</sub><sup>2</sup>/<i>σ</i><sub>2</sub><sup>2</sup>, could now be approximated by <a href="/wiki/Student%27s_t_distribution" class="mw-redirect" title="Student's t distribution">Student's t distribution</a> with these <i>ν</i> degrees of freedom. But this <i>ν</i> contains the population variances <i>σ<sub>i</sub></i><sup>2</sup>, and these are unknown. The following estimate only replaces the population variances by the sample variances: </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 {\nu }}\approx {\frac {(g_{1}+g_{2})^{2}}{g_{1}^{2}/(n_{1}-1)+g_{2}^{2}/(n_{2}-1)}}\quad {\text{ where }}g_{i}=s_{i}^{2}/n_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\nu }}\approx {\frac {(g_{1}+g_{2})^{2}}{g_{1}^{2}/(n_{1}-1)+g_{2}^{2}/(n_{2}-1)}}\quad {\text{ where }}g_{i}=s_{i}^{2}/n_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9aa6f7b79a933eb176177e5e2b004afb3df26e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:51.87ex; height:6.843ex;" alt="{\displaystyle {\hat {\nu }}\approx {\frac {(g_{1}+g_{2})^{2}}{g_{1}^{2}/(n_{1}-1)+g_{2}^{2}/(n_{2}-1)}}\quad {\text{ where }}g_{i}=s_{i}^{2}/n_{i}.}"></span></dd></dl> <p>This <span class="mwe-math-element"><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 {\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8c4f0785c6b4c01435dcc0aa5b9cfba84bb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.316ex; height:2.176ex;" alt="{\displaystyle {\hat {\nu }}}"></span> is a random variable. A t distribution with a random number of degrees of freedom does not exist. Nevertheless, the Behrens–Fisher <i>T</i> can be compared with a corresponding quantile of <a href="/wiki/Student%27s_t_distribution" class="mw-redirect" title="Student's t distribution">Student's t distribution</a> with these estimated numbers of degrees of freedom, <span class="mwe-math-element"><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 {\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8c4f0785c6b4c01435dcc0aa5b9cfba84bb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.316ex; height:2.176ex;" alt="{\displaystyle {\hat {\nu }}}"></span>, which is generally non-integer. In this way, the boundary between acceptance and rejection region of the test statistic <i>T</i> is calculated based on the empirical variances <i>s<sub>i</sub></i><sup>2</sup>, in a way that is a smooth function of these. </p><p>This method also does not give exactly the nominal rate, but is generally not too far off.<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. (September 2010)">citation needed</span></a></i>]</sup> However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it is more accurate to use <a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's t-test</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2010)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_approaches">Other approaches</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=7" title="Edit section: Other approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of different approaches to the general problem have been proposed, some of which claim to "solve" some version of the problem. Among these are,<sup id="cite_ref-DMMS_7-0" class="reference"><a href="#cite_note-DMMS-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><ul><li>that of Chapman in 1950,<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></li> <li>that of Prokof’yev and Shishkin in 1974,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>that of Dudewicz and Ahmed in 1998.<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></li> <li>that of Chang Wang in 2022.<sup id="cite_ref-:0_11-0" class="reference"><a href="#cite_note-:0-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <p>In Dudewicz’s comparison of selected methods,<sup id="cite_ref-DMMS_7-1" class="reference"><a href="#cite_note-DMMS-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> it was found that the Dudewicz–Ahmed procedure is recommended for practical use. </p> <div class="mw-heading mw-heading3"><h3 id="Exact_solutions_to_the_common_and_generalized_Behrens–Fisher_problems"><span id="Exact_solutions_to_the_common_and_generalized_Behrens.E2.80.93Fisher_problems"></span>Exact solutions to the common and generalized Behrens–Fisher problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=8" title="Edit section: Exact solutions to the common and generalized Behrens–Fisher problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For several decades, it was commonly believed that no exact solution to the common Behrens–Fisher problem existed.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Assertions of "commonly believed" should be backed up. (May 2020)">citation needed</span></a></i>]</sup> However, it was proved in 1966 that it has an exact solution.<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> In 2018 the probability density function of a generalized Behrens–Fisher distribution of <i>m</i> means and <i>m</i> distinct standard errors from <i>m</i> samples of distinct sizes from independent normal distributions with distinct means and variances was proved and the paper also examined its asymptotic approximations.<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> A follow-up paper showed that the classic paired <i>t</i>-test is a central Behrens–Fisher problem with a non-zero population correlation coefficient and derived its corresponding probability density function by solving its associated non-central Behrens–Fisher problem with a nonzero population correlation coefficient.<sup id="cite_ref-Xiao2018b_14-0" class="reference"><a href="#cite_note-Xiao2018b-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It also solved a more general non-central Behrens–Fisher problem with a non-zero population correlation coefficient in the appendix.<sup id="cite_ref-Xiao2018b_14-1" class="reference"><a href="#cite_note-Xiao2018b-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Variants">Variants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=9" title="Edit section: Variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A minor variant of the Behrens–Fisher problem has been studied.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In this instance the problem is, assuming that the two population-means are in fact the same, to make inferences about the common mean: for example, one could require a <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for the common mean. </p> <div class="mw-heading mw-heading2"><h2 id="Generalisations">Generalisations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Behrens%E2%80%93Fisher_problem&action=edit&section=10" title="Edit section: Generalisations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One generalisation of the problem involves <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distributions</a> with unknown covariance matrices, and is known as the <a href="/wiki/Multivariate_Behrens%E2%80%93Fisher_problem" title="Multivariate Behrens–Fisher problem">multivariate Behrens–Fisher problem</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">nonparametric</a> Behrens–Fisher problem does not assume that the distributions are normal.<sup id="cite_ref-Brunner2000_17-0" class="reference"><a href="#cite_note-Brunner2000-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nparcomp_18-0" class="reference"><a href="#cite_note-nparcomp-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Tests include the <a href="/wiki/Cucconi_test" title="Cucconi test">Cucconi test</a> of 1968 and the <a href="/wiki/Lepage_test" title="Lepage test">Lepage test</a> of 1971. </p> <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=Behrens%E2%80%93Fisher_problem&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><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">Lehmann (1975) p.95</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">Lehmann (1975) Section 7</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"><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="CITEREFFisher1935" class="citation journal cs1">Fisher, R. A. (1935). "The fiducial argument in statistical inference". <i>Annals of Eugenics</i>. <b>8</b> (4): 391–398. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-1809.1935.tb02120.x">10.1111/j.1469-1809.1935.tb02120.x</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2440%2F15222">2440/15222</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Eugenics&rft.atitle=The+fiducial+argument+in+statistical+inference&rft.volume=8&rft.issue=4&rft.pages=391-398&rft.date=1935&rft_id=info%3Ahdl%2F2440%2F15222&rft_id=info%3Adoi%2F10.1111%2Fj.1469-1809.1935.tb02120.x&rft.aulast=Fisher&rft.aufirst=R.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cmu.edu/dietrich/philosophy/docs/seidenfeld/Fishers%20Fiducial%20Argument%20and%20Bayes%20Theorem.pdf">"R. A. Fisher's Fiducial Argument and Bayes' Theorem by Teddy Seidenfeld"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=R.+A.+Fisher%27s+Fiducial+Argument+and+Bayes%27+Theorem+by+Teddy+Seidenfeld&rft_id=https%3A%2F%2Fwww.cmu.edu%2Fdietrich%2Fphilosophy%2Fdocs%2Fseidenfeld%2FFishers%2520Fiducial%2520Argument%2520and%2520Bayes%2520Theorem.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/313652727">"Sezer, A. et al. Comparison of confidence intervals for the Behrens–Fisher Problem <i>Comm. Stats.</i> 2015"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sezer%2C+A.+et+al.+Comparison+of+confidence+intervals+for+the+Behrens%E2%80%93Fisher+Problem+Comm.+Stats.+2015&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F313652727&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></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">Welch (1938, 1947)</span> </li> <li id="cite_note-DMMS-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-DMMS_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DMMS_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Dudewicz, Ma, Mai, and Su (2007)</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChapman1950" class="citation journal cs1">Chapman, D. G. (1950). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177729755">"Some two sample tests"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>21</b> (4): 601–606. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177729755">10.1214/aoms/1177729755</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Some+two+sample+tests&rft.volume=21&rft.issue=4&rft.pages=601-606&rft.date=1950&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177729755&rft.aulast=Chapman&rft.aufirst=D.+G.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177729755&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProkof'yevShishkin1974" class="citation journal cs1">Prokof'yev, V. N.; Shishkin, A. D. (1974). "Successive classification of normal sets with unknown variances". <i>Radio Engng. Electron. Phys</i>. <b>19</b> (2): 141–143.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Radio+Engng.+Electron.+Phys&rft.atitle=Successive+classification+of+normal+sets+with+unknown+variances&rft.volume=19&rft.issue=2&rft.pages=141-143&rft.date=1974&rft.aulast=Prokof%27yev&rft.aufirst=V.+N.&rft.au=Shishkin%2C+A.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Dudewicz & Ahmed (1998, 1999)</span> </li> <li id="cite_note-:0-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang2022" class="citation arxiv cs1">Wang, Chang (2022). "A New Non-asymptotic t-test for Behrens-Fisher Problems". <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/2210.16473">2210.16473</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.ST">math.ST</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+New+Non-asymptotic+t-test+for+Behrens-Fisher+Problems&rft.date=2022&rft_id=info%3Aarxiv%2F2210.16473&rft.aulast=Wang&rft.aufirst=Chang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKabe1966" class="citation journal cs1">Kabe, D. G. (December 1966). "On the exact distribution of the Fisher-Behren'-Welch statistic". <i>Metrika</i>. <b>10</b> (1): 13–15. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02613414">10.1007/BF02613414</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120965543">120965543</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrika&rft.atitle=On+the+exact+distribution+of+the+Fisher-Behren%27-Welch+statistic&rft.volume=10&rft.issue=1&rft.pages=13-15&rft.date=1966-12&rft_id=info%3Adoi%2F10.1007%2FBF02613414&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120965543%23id-name%3DS2CID&rft.aulast=Kabe&rft.aufirst=D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXiao2018" class="citation journal cs1">Xiao, Yongshun (22 March 2018). <a rel="nofollow" class="external text" href="http://www.pphmj.com/abstract/11558.htm">"On the Solution of a Generalized Behrens-Fisher Problem"</a>. <i>Far East Journal of Theoretical Statistics</i>. <b>54</b> (1): 21–140. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17654%2FTS054010021">10.17654/TS054010021</a><span class="reference-accessdate">. Retrieved <span class="nowrap">21 May</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Far+East+Journal+of+Theoretical+Statistics&rft.atitle=On+the+Solution+of+a+Generalized+Behrens-Fisher+Problem&rft.volume=54&rft.issue=1&rft.pages=21-140&rft.date=2018-03-22&rft_id=info%3Adoi%2F10.17654%2FTS054010021&rft.aulast=Xiao&rft.aufirst=Yongshun&rft_id=http%3A%2F%2Fwww.pphmj.com%2Fabstract%2F11558.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-Xiao2018b-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Xiao2018b_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Xiao2018b_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXiao2018" class="citation journal cs1">Xiao, Yongshun (12 December 2018). <a rel="nofollow" class="external text" href="http://www.pphmj.com/abstract/12204.htm">"On the Solution of a Non-Central Behrens-Fisher Problem with a Non-Zero Population Correlation Coefficient"</a>. <i>Far East Journal of Theoretical Statistics</i>. <b>54</b> (6): 527–600. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17654%2FTS054060527">10.17654/TS054060527</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125245802">125245802</a><span class="reference-accessdate">. Retrieved <span class="nowrap">21 May</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Far+East+Journal+of+Theoretical+Statistics&rft.atitle=On+the+Solution+of+a+Non-Central+Behrens-Fisher+Problem+with+a+Non-Zero+Population+Correlation+Coefficient&rft.volume=54&rft.issue=6&rft.pages=527-600&rft.date=2018-12-12&rft_id=info%3Adoi%2F10.17654%2FTS054060527&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125245802%23id-name%3DS2CID&rft.aulast=Xiao&rft.aufirst=Yongshun&rft_id=http%3A%2F%2Fwww.pphmj.com%2Fabstract%2F12204.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Young, G. A., Smith, R. L. (2005) <i>Essentials of Statistical Inference</i>, CUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83971-8" title="Special:BookSources/0-521-83971-8">0-521-83971-8</a> (page 204)</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Belloni & Didier (2008)</span> </li> <li id="cite_note-Brunner2000-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brunner2000_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrunner2000" class="citation journal cs1">Brunner, E. (2000). "Nonparametric Behrens–Fisher Problem: Asymptotic Theory and a Small Sample Approximation". <i>Biometrical Journal</i>. <b>42</b>: 17–25. <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%2F%28SICI%291521-4036%28200001%2942%3A1%3C17%3A%3AAID-BIMJ17%3E3.0.CO%3B2-U">10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrical+Journal&rft.atitle=Nonparametric+Behrens%E2%80%93Fisher+Problem%3A+Asymptotic+Theory+and+a+Small+Sample+Approximation&rft.volume=42&rft.pages=17-25&rft.date=2000&rft_id=info%3Adoi%2F10.1002%2F%28SICI%291521-4036%28200001%2942%3A1%3C17%3A%3AAID-BIMJ17%3E3.0.CO%3B2-U&rft.aulast=Brunner&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> <li id="cite_note-nparcomp-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-nparcomp_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKonietschke2015" class="citation journal cs1">Konietschke, Frank (2015). <a rel="nofollow" class="external text" href="https://www.jstatsoft.org/article/view/v064i09">"nparcomp: An R Software Package for Nonparametric Multiple Comparisons and Simultaneous Confidence Intervals"</a>. <i>Journal of Statistical Software</i>. <b>64</b> (9). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v064.i09">10.18637/jss.v064.i09</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">26 September</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Software&rft.atitle=nparcomp%3A+An+R+Software+Package+for+Nonparametric+Multiple+Comparisons+and+Simultaneous+Confidence+Intervals&rft.volume=64&rft.issue=9&rft.date=2015&rft_id=info%3Adoi%2F10.18637%2Fjss.v064.i09&rft.aulast=Konietschke&rft.aufirst=Frank&rft_id=https%3A%2F%2Fwww.jstatsoft.org%2Farticle%2Fview%2Fv064i09&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></span> </li> </ol></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-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">February 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> <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=Behrens%E2%80%93Fisher_problem&action=edit&section=12" 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="CITEREFBehrens1929" class="citation journal cs1"><a href="/w/index.php?title=Walter_Ulrich_Behrens&action=edit&redlink=1" class="new" title="Walter Ulrich Behrens (page does not exist)">Behrens, W. U.</a> (1929). <a rel="nofollow" class="external text" href="http://catalog.hathitrust.org/Record/007924569">"Ein Beitrag zur Fehlerberechnung bei wenigen Beobachtungen"</a> [A contribution to error estimation with few observations]. <i>Landwirtschaftliche Jahrbücher</i>. <b>68</b>. Berlin: Wiegandt and Hempel: 807–37.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Landwirtschaftliche+Jahrb%C3%BCcher&rft.atitle=Ein+Beitrag+zur+Fehlerberechnung+bei+wenigen+Beobachtungen&rft.volume=68&rft.pages=807-37&rft.date=1929&rft.aulast=Behrens&rft.aufirst=W.+U.&rft_id=http%3A%2F%2Fcatalog.hathitrust.org%2FRecord%2F007924569&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelloniDidier2008" class="citation journal cs1">Belloni, A.; Didier, G. (2008). "On the Behrens–Fisher Problem: A Globally Convergent Algorithm and a Finite-Sample Study of the Wald, LR and LM Tests". <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a></i>. <b>36</b> (5): 2377–2408. <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/0811.0672">0811.0672</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F07-AOS528">10.1214/07-AOS528</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15968707">15968707</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=On+the+Behrens%E2%80%93Fisher+Problem%3A+A+Globally+Convergent+Algorithm+and+a+Finite-Sample+Study+of+the+Wald%2C+LR+and+LM+Tests&rft.volume=36&rft.issue=5&rft.pages=2377-2408&rft.date=2008&rft_id=info%3Aarxiv%2F0811.0672&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15968707%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1214%2F07-AOS528&rft.aulast=Belloni&rft.aufirst=A.&rft.au=Didier%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChangPal2008" class="citation journal cs1">Chang, CH; Pal, N (2008). "A revisit to the Behrens–Fisher problem: Comparison of five test methods". <i>Communications in Statistics - Simulation and Computation</i>. <b>37</b> (6): 1064–1085. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610910802049599">10.1080/03610910802049599</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:32811488">32811488</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Simulation+and+Computation&rft.atitle=A+revisit+to+the+Behrens%E2%80%93Fisher+problem%3A+Comparison+of+five+test+methods&rft.volume=37&rft.issue=6&rft.pages=1064-1085&rft.date=2008&rft_id=info%3Adoi%2F10.1080%2F03610910802049599&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32811488%23id-name%3DS2CID&rft.aulast=Chang&rft.aufirst=CH&rft.au=Pal%2C+N&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudewiczAhmed1998" class="citation journal cs1">Dudewicz, E. J.; Ahmed, S. U. (1998). "New exact and asymptotically optimal solution to the Behrens–Fisher problem, with tables". <i>American Journal of Mathematical and Management Sciences</i>. <b>18</b> (3–4): 359–426. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01966324.1998.10737471">10.1080/01966324.1998.10737471</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematical+and+Management+Sciences&rft.atitle=New+exact+and+asymptotically+optimal+solution+to+the+Behrens%E2%80%93Fisher+problem%2C+with+tables&rft.volume=18&rft.issue=3%E2%80%934&rft.pages=359-426&rft.date=1998&rft_id=info%3Adoi%2F10.1080%2F01966324.1998.10737471&rft.aulast=Dudewicz&rft.aufirst=E.+J.&rft.au=Ahmed%2C+S.+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudewiczAhmed1999" class="citation journal cs1">Dudewicz, E. J.; Ahmed, S. U. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01966324.1999.10737478">"New exact and asymptotically optimal heteroscedastic statistical procedures and tables, II"</a>. <i>American Journal of Mathematical and Management Sciences</i>. <b>19</b> (1–2): 157–180. <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/10.1080%2F01966324.1999.10737478">10.1080/01966324.1999.10737478</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematical+and+Management+Sciences&rft.atitle=New+exact+and+asymptotically+optimal+heteroscedastic+statistical+procedures+and+tables%2C+II.&rft.volume=19&rft.issue=1%E2%80%932&rft.pages=157-180&rft.date=1999&rft_id=info%3Adoi%2F10.1080%2F01966324.1999.10737478&rft.aulast=Dudewicz&rft.aufirst=E.+J.&rft.au=Ahmed%2C+S.+U.&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%252F01966324.1999.10737478&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudewiczMaMaiSu2007" class="citation journal cs1">Dudewicz, E. J.; Ma, Y.; Mai, S. E.; Su, H. (2007). "Exact solutions to the Behrens–Fisher problem: Asymptotically optimal and finite sample efficient choice among". <i>Journal of Statistical Planning and Inference</i>. <b>137</b> (5): 1584–1605. <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.2006.09.007">10.1016/j.jspi.2006.09.007</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=Exact+solutions+to+the+Behrens%E2%80%93Fisher+problem%3A+Asymptotically+optimal+and+finite+sample+efficient+choice+among&rft.volume=137&rft.issue=5&rft.pages=1584-1605&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2006.09.007&rft.aulast=Dudewicz&rft.aufirst=E.+J.&rft.au=Ma%2C+Y.&rft.au=Mai%2C+S.+E.&rft.au=Su%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFisher1935" class="citation journal cs1">Fisher, R. A. (1935). "The fiducial argument in statistical inference". <i>Annals of Eugenics</i>. <b>8</b> (4): 391–398. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-1809.1935.tb02120.x">10.1111/j.1469-1809.1935.tb02120.x</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2440%2F15222">2440/15222</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Eugenics&rft.atitle=The+fiducial+argument+in+statistical+inference&rft.volume=8&rft.issue=4&rft.pages=391-398&rft.date=1935&rft_id=info%3Ahdl%2F2440%2F15222&rft_id=info%3Adoi%2F10.1111%2Fj.1469-1809.1935.tb02120.x&rft.aulast=Fisher&rft.aufirst=R.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFisher1941" class="citation journal cs1">Fisher, R. A. (1941). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-1809.1941.tb02281.x">"The Asymptotic Approach to Behrens' Integral with further Tables for the d Test of Significance"</a>. <i>Annals of Eugenics</i>. <b>11</b>: 141–172. <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/10.1111%2Fj.1469-1809.1941.tb02281.x">10.1111/j.1469-1809.1941.tb02281.x</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Eugenics&rft.atitle=The+Asymptotic+Approach+to+Behrens%27+Integral+with+further+Tables+for+the+d+Test+of+Significance&rft.volume=11&rft.pages=141-172&rft.date=1941&rft_id=info%3Adoi%2F10.1111%2Fj.1469-1809.1941.tb02281.x&rft.aulast=Fisher&rft.aufirst=R.+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1469-1809.1941.tb02281.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraserRousseau2008" class="citation journal cs1">Fraser, D. A. S.; <a href="/wiki/Judith_Rousseau" title="Judith Rousseau">Rousseau, J.</a> (2008). <a rel="nofollow" class="external text" href="https://basepub.dauphine.fr/handle/123456789/555">"Studentization and deriving accurate p-values"</a>. <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>. <b>95</b> (1): 1–16. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2Fasm093">10.1093/biomet/asm093</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Studentization+and+deriving+accurate+p-values&rft.volume=95&rft.issue=1&rft.pages=1-16&rft.date=2008&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2Fasm093&rft.aulast=Fraser&rft.aufirst=D.+A.+S.&rft.au=Rousseau%2C+J.&rft_id=https%3A%2F%2Fbasepub.dauphine.fr%2Fhandle%2F123456789%2F555&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li>Lehmann, E. L. (1975) <i>Nonparametrics: Statistical Methods Based on Ranks</i>, Holden-Day <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8162-4996-6" title="Special:BookSources/0-8162-4996-6">0-8162-4996-6</a>, McGraw-Hill <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-037073-7" title="Special:BookSources/0-07-037073-7">0-07-037073-7</a></li> <li><a href="/w/index.php?title=Harold_Ruben&action=edit&redlink=1" class="new" title="Harold Ruben (page does not exist)">Ruben, H.</a> (2002)<a rel="nofollow" class="external text" href="https://web.archive.org/web/20070313091422/http://sankhya.isical.ac.in/search/servlet/SSearch?s_order=2&choice1=author&text1=Ruben&opt1=And&choice2=title&text2=&opt2=And&choice3=title&text3=&opt3=And&choice4=keyword&text4=&rel_yr=equalto&yearsrch=2002&rel_vol=equalto&volumesrch=64&series=on&part=on&amssrch=&num=20&cntr=0">"A simple conservative and robust solution of the Behrens–Fisher problem"</a>, <i><a href="/wiki/Sankhya_(journal)" title="Sankhya (journal)">Sankhyā:The Indian Journal of Statistics</a></i>, Series A, 64 (1),139–155.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPardoPardo2007" class="citation journal cs1">Pardo, JA; Pardo, MD (2007). "A simulation study of a new family of test statistics for the Behrens–Fisher problem". <i>Kybernetes</i>. <b>36</b> (5–6): 806–816. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1108%2F03684920710749866">10.1108/03684920710749866</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Kybernetes&rft.atitle=A+simulation+study+of+a+new+family+of+test+statistics+for+the+Behrens%E2%80%93Fisher+problem&rft.volume=36&rft.issue=5%E2%80%936&rft.pages=806-816&rft.date=2007&rft_id=info%3Adoi%2F10.1108%2F03684920710749866&rft.aulast=Pardo&rft.aufirst=JA&rft.au=Pardo%2C+MD&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawilowsky2002" class="citation journal cs1">Sawilowsky, Shlomo S (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1036109940">"Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ<sub>1</sub> ≠ σ<sub>2</sub>"</a>. <i>Journal of Modern Applied Statistical Methods</i>. <b>1</b> (2). <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/10.22237%2Fjmasm%2F1036109940">10.22237/jmasm/1036109940</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+Modern+Applied+Statistical+Methods&rft.atitle=Fermat%2C+Schubert%2C+Einstein%2C+and+Behrens%E2%80%93Fisher%3A+The+Probable+Difference+Between+Two+Means+When+%CF%83%3Csub%3E1%3C%2Fsub%3E+%E2%89%A0+%CF%83%3Csub%3E2%3C%2Fsub%3E&rft.volume=1&rft.issue=2&rft.date=2002&rft_id=info%3Adoi%2F10.22237%2Fjmasm%2F1036109940&rft.aulast=Sawilowsky&rft.aufirst=Shlomo+S&rft_id=https%3A%2F%2Fdoi.org%2F10.22237%252Fjmasm%252F1036109940&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWelch1938" class="citation journal cs1">Welch, B. L. (1938). "The significance of the difference between two means when the population variances are unequal". <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>. <b>29</b> (3/4): 350–62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2332010">10.2307/2332010</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2332010">2332010</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=The+significance+of+the+difference+between+two+means+when+the+population+variances+are+unequal&rft.volume=29&rft.issue=3%2F4&rft.pages=350-62&rft.date=1938&rft_id=info%3Adoi%2F10.2307%2F2332010&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2332010%23id-name%3DJSTOR&rft.aulast=Welch&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWelch1947" class="citation cs2">Welch, B. L. (1947), "The generalization of "Student's" problem when several different population variances are involved", <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>, <b>34</b> (1–2): 28–35, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F34.1-2.28">10.1093/biomet/34.1-2.28</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=0019277">0019277</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20287819">20287819</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=The+generalization+of+%22Student%27s%22+problem+when+several+different+population+variances+are+involved&rft.volume=34&rft.issue=1%E2%80%932&rft.pages=28-35&rft.date=1947&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D19277%23id-name%3DMR&rft_id=info%3Apmid%2F20287819&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F34.1-2.28&rft.aulast=Welch&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoinovNikulin1995" class="citation journal cs1">Voinov, V.; Nikulin, M. (1995). "On the problem of means of weighted normal populations". <i>Questiio</i>. <b>19</b> (2): 7–20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Questiio&rft.atitle=On+the+problem+of+means+of+weighted+normal+populations&rft.volume=19&rft.issue=2&rft.pages=7-20&rft.date=1995&rft.aulast=Voinov&rft.aufirst=V.&rft.au=Nikulin%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhengShiMa2010" class="citation journal cs1">Zheng, SR; Shi, NZ; Ma, WQ (2010). "Statistical inference on difference or ratio of means from heteroscedastic normal populations". <i>Journal of Statistical Planning and Inference</i>. <b>140</b> (5): 1236–1242. <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.2009.11.010">10.1016/j.jspi.2009.11.010</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=Statistical+inference+on+difference+or+ratio+of+means+from+heteroscedastic+normal+populations&rft.volume=140&rft.issue=5&rft.pages=1236-1242&rft.date=2010&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2009.11.010&rft.aulast=Zheng&rft.aufirst=SR&rft.au=Shi%2C+NZ&rft.au=Ma%2C+WQ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABehrens%E2%80%93Fisher+problem" 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=Behrens%E2%80%93Fisher_problem&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Dong, B.L. (2004) <a rel="nofollow" class="external text" href="http://web.uvic.ca/econ/research/papers/pdfs/ewp0404.pdf">The Behrens–Fisher Problem: An Empirical Likelihood Approach</a> Econometrics Working Paper EWP0404, University of Victoria</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Behrens–Fisher_problem&oldid=1216557215">https://en.wikipedia.org/w/index.php?title=Behrens–Fisher_problem&oldid=1216557215</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematical_problems" title="Category:Mathematical problems">Mathematical problems</a></li><li><a href="/wiki/Category:Statistical_hypothesis_testing" title="Category:Statistical hypothesis testing">Statistical hypothesis testing</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_July_2014" title="Category:Articles with unsourced statements from July 2014">Articles with unsourced statements from July 2014</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2010" title="Category:Articles with unsourced statements from September 2010">Articles with unsourced statements from September 2010</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2020" title="Category:Articles with unsourced statements from May 2020">Articles with unsourced statements from May 2020</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_February_2010" title="Category:Articles lacking in-text citations from February 2010">Articles lacking in-text citations from February 2010</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Pages_with_listed_invalid_ISBNs" title="Category:Pages with listed invalid ISBNs">Pages with listed invalid ISBNs</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 March 2024, at 18:34<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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