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

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  • Wolf-Dieter Richter

    (Institute of Mathematics, University of Rostock, 18057 Rostock, Germany)

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.

Suggested Citation

  • Wolf-Dieter Richter, 2020. "Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic," Stats, MDPI, vol. 3(3), pages 1-13, August.
    • Handle: RePEc:gam:jstats:v:3:y:2020:i:3:p:21-342:d:403827
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    References listed on IDEAS

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    1. Maity, Arnab & Sherman, Michael, 2006. "The Two-Sample T Test With One Variance Unknown," The American Statistician, American Statistical Association, vol. 60, pages 163-166, May.
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