On the two-sample Behrens–Fisher problem for high-dimensional data

Abstract

In this paper, we study the limiting distribution of Chen-Qin’s test statistic and propose a novel weighted bootstrap test procedure for the high-dimensional two-sample Behrens–Fisher problem. We first show that the test statistic has an asymptotic null that is a mixture of a chi-square-type mixture distribution and a normal distribution, without imposing either the normal assumption or a factor-like model assumption on the underlying distributions. To gain insight into the asymptotic null distribution of the test statistic, we show that under stronger restrictions on the covariance matrices and the null hypothesis, the test statistic is either asymptotically normal or a chi-square-type mixture distribution. The power properties of the test are evaluated asymptotically under the high-dimensional local and fixed alternative hypothesis. We also derive that the proposed weighted bootstrap test procedure has correct test level asymptotically. Two simulation studies and a real data example show that the new weighted bootstrap procedure significantly outperforms other benchmarks in terms of size control and is comparable in terms of power.