Testing homogeneity of mean vectors under heteroscedasticity in high-dimension

This paper is concerned with the problem of testing the homogeneity of mean vectors. The testing problem is without assuming common covariance matrix. We proposed a testing statistic based on the variation matrix due to the hypothesis and the unbiased estimator of the covariance matrix. The limiting null and non-null distributions are derived as each sample size and the dimensionality go to infinity together under a general population distribution, which includes elliptical distribution with finite fourth moments or distribution assumed in Chen and Qin (2010). In two-sample case, our proposed test has the same asymptotic power as Chen and Qin (2010)’s test. In addition, it is found that our proposed test has the same asymptotic power as the one of Dempster’s trace statistic for MANOVA proposed in Fujikoshi et al. (2004) for the case that the population distributions are multivariate normal with common covariance matrix for all groups. A small scale simulation study is performed to compare the actual error probability of the first kind with the nominal.