文章基本信息

标题：Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic
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作者：Wolf-Dieter Richter
期刊名称：Stats
电子版ISSN：2571-905X
出版年度：2020
卷号：3
期号：3
页码：330-342
DOI：10.3390/stats3030021
出版社：MDPI AG
摘要：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.
关键词：heteroscedasticity; unbalancedness; sums of weighted chi-squares; variance ratio; Welch approximation; three pillar bridges property