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This paper is concerned with the problem of comparing the population means oftwo groups of independent observations. An approximate randomization testprocedure based on the test statistic of Chen and Qin (2010) is proposed. Theasymptotic behavior of the test statistic as well as the randomized statisticis studied under weak conditions. In our theoretical framework, observationsare not assumed to be identically distributed even within groups. No conditionon the eigenstructure of the covariance matrices is imposed. And the samplesizes of the two groups are allowed to be unbalanced. Under general conditions,all possible asymptotic distributions of the test statistic are obtained. Wederive the asymptotic level and local power of the approximate randomizationtest procedure. Our theoretical results show that the proposed test procedurecan adapt to all possible asymptotic distributions of the test statistic andalways has correct test level asymptotically. Also, the proposed test procedurehas good power behavior. Our numerical experiments show that the proposed testprocedure has favorable performance compared with several alternative testprocedures.

An approximate randomization test for high-dimensional two-sample Behrens-Fisher problem under arbitrary covariances