Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

Wolf-Dieter Richter
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Wolf-Dieter Richter: Institute of Mathematics, University of Rostock, 18057 Rostock, Germany

Stats, 2020, vol. 3, issue 3, 1-13

Abstract: We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic.

Keywords: heteroscedasticity; unbalancedness; sums of weighted chi-squares; variance ratio; Welch approximation; three pillar bridges property (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2020
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