Behrens-Fisher Problem
The Behrens-Fisher problem is an influential statistical problem that was first posed by German statistician Ernst Behrens in 1908 and later extended by English statistician R.A. Fisher in 1925. The problem is concerned with determining whether the means of two independent, normally distributed populations are significantly different. The Behrens-Fisher problem is an important fundamental problem in statistics and has been the topic of numerous studies over the past century.
Definition
The Behrens-Fisher problem is a statistical inference problem in which the goal is to determine whether the means of two independent, normally distributed populations differ significantly. Specifically, a statistician must make a decision concerning whether the difference between the two means is large enough to reject the null hypothesis that the population means are equal.
History
The Behrens-Fisher problem was first posed by German statistician Ernst Behrens in 1908. Behrens proposed the use of the Student’s t-distribution, which had been developed by William Gosset, in order to test for the difference between the two means. However, the t-distribution only works in the case of a single population. R.A. Fisher extended this problem in 1925 to the case of two independent populations and proposed the use of the F-distribution to test for the difference between the two means. The F-distribution is a generalization of the t-distribution which takes into account the variance of each population.
References
Behrens, E. (1908). Um die Beurteilung mathematischer Ergebnisse. Journal für die Reine und Angewandte Mathematik, 134, 219-220.
Fisher, R. A. (1925). Theory of statistical estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 22, 700-725.
Gosset, W. S. (1908). The probable error of a mean. Biometrika, 6, 1-25.