In statistics, the Behrens–Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples. The Behrens–Fisher Problem has been solved, in fact there are three solutions: that of Chapman in 1950, that of Prokof'yev and Shishkin in 1974, and that of Dudewicz and Ahmed in 1998.

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  • In statistics, the Behrens–Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples. The Behrens–Fisher Problem has been solved, in fact there are three solutions: that of Chapman in 1950, that of Prokof'yev and Shishkin in 1974, and that of Dudewicz and Ahmed in 1998. The solutions have been compared by Dudewicz, Ma, Mai, and Su in 2007, and it was found that the Dudewicz-Ahmed procedure is recommended for practical use. Ronald Fisher in 1935 introduced fiducial inference in order to apply it to this problem. He referred to an earlier paper by W. V. Behrens from 1929. Behrens and Fisher proposed to find the probability distribution of <math> T \equiv {\bar x_1 - \bar x_2 \over \sqrt{s_1^2/n_1 + s_2^2/n_2}} </math> where <math> \bar x_1 </math> and <math> \bar x_2 </math> are the two sample means, and <math> s_1 </math> and <math> s_2 </math> are their standard deviations. Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations, <math> {s_1 / \sqrt{n_1} \over \sqrt{s_1^2/n_1 + s_2^2/n_2}}. </math> Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be rejected with probability α if the means were in fact equal. Many other methods of treating the problem have been proposed since.
  • Das Behrens-Fisher-Problem ist eine Problemstellung der mathematischen Statistik, deren exakte Lösungen nachgewiesenermaßen unerwünschte Eigenschaften haben, weswegen man Approximationen bevorzugt. Gesucht ist ein nichtrandomisierter ähnlicher Test der Nullhypothese gleicher Erwartungswerte, <math>H_0\colon\,\mu_1=\mu_2</math>, zweier normalverteilter Grundgesamtheiten, deren Varianzen <math>\sigma_1^2</math> und <math>\sigma_2^2</math> unbekannt sind und nicht als gleich vorausgesetzt werden. Die Ähnlichkeit des Tests besagt dabei, dass die Nullhypothese bei deren Gültigkeit exakt mit Wahrscheinlichkeit <math>\,\alpha</math>, dem vorgegebenen Signifikanzniveau, abgelehnt wird, wie groß und unterschiedlich auch immer die unbekannten Varianzen <math>\sigma_1^2</math> und <math>\sigma_2^2</math> sind. Aus Gründen der Macht des Tests bezieht man sich auf folgende „Behrens-Fisher“-Testgröße: <math> T = {\bar x_1 - \bar x_2 \over \sqrt{s_1^2/n_1 + s_2^2/n_2}},</math> wobei <math>\bar x_1</math> und <math>\bar x_2</math> die Mittelwerte und <math>\,s_1</math> und <math>\,s_2</math> die Standardabweichungen der beiden Stichproben sind; mit <math>\,n_1</math> und <math>\,n_2</math> wird deren jeweiliger Umfang bezeichnet. Das Behrens-Fisher-Problem verallgemeinert den t-Test für zwei unabhängige Stichproben; dieser setzt nämlich voraus, dass die Varianzen beider Grundgesamtheiten übereinstimmen.
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dbpprop:unsolvedProperty
  • Only approximate solutions are known
  • Statistics
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  • In statistics, the Behrens–Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples. The Behrens–Fisher Problem has been solved, in fact there are three solutions: that of Chapman in 1950, that of Prokof'yev and Shishkin in 1974, and that of Dudewicz and Ahmed in 1998.
  • Das Behrens-Fisher-Problem ist eine Problemstellung der mathematischen Statistik, deren exakte Lösungen nachgewiesenermaßen unerwünschte Eigenschaften haben, weswegen man Approximationen bevorzugt.
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  • Behrens–Fisher problem
  • Behrens-Fisher-Problem
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