New Exact and Asymptotically Optimal Solution to the Behrens-Fisher Problem, with Tables

SYNOPTIC ABSTRACT

The problem of testing equality of two means when the variances are not known has been called the most important problem of applied statistics (H. Scheffé). It has been proven that, under certain assumptions, this problem has no solution (Linnik). Thus, for most of the 20th century statisticians have been developing approximate solutions to this important problem (Welch, Chernoff, and others, and this line of research has been continuing). However, there are two exact solutions: those of Chapman, and of Prokof'yev and Shishkin (and those were compared by the first author and Taneja). In this paper we give a new exact solution to the Behrens-Fisher Problem. This new solution is also asymptotically optimal, and thus is superior to all other existing exact solutions. (Of course, the three exact solutions are all superior to the many approximate solutions–why would anyone use an approximate solution when there are exact answers?) Carefully constructed tables needed for implementation are given, and cover a very wide range of values of the parameters.

Key Words and Phrases:

• Behrens-Fisher Problem
• Tests of Hypotheses
• Heteroscedasticity
• Equality of Means Testing
• Exact Level Tests
• Asymptotically Optimal Tests