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

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.