Solving Some Behrens-Fisher Problems Using Modified Bartlett Correction

Abstract

The Behrens-Fisher (BF) problems refer to compare the means or mean vectors of several normal populations without assuming the equality of the variances or covariance matrices of those normal populations. These BF problems are challenging and caught much attention for decades since the standard testing procedures such as the t-test, F-test, Hotelling T2-test, or the Lawley-Hotelling trace test may fail for these BF problems.

In this thesis, we solve various BF problems by applying the modified Bartlett correction of Fujikoshi (2000). These BF problems include heterogenous one-way ANOVA, multi-way ANOVA, one-way MANOVA, two-way MANOVA, and regression coefficient comparison under heteroscedasticity. For each BF problem, we show that the asymptotic distribution of the test statistic is Chi-square with some known degrees of freedom and we find out the expressions of the asymptotic mean and variance of the test statistic which allow us to apply the modified Bartlett correction. In each of these BF problems, by simulation studies and real data applications, we find that the resulting modified Bartlett test works well compared with the existing approximate solutions to the associated Behrens-Fisher problem.