Tests For Mean Vectors In High Dimensional K-sample With Unequal Covariance Matrices

In this dissertation, the problems of testing hypotheses on mean vectors are studiedin (ultra-)high dimensional data with unequal covariance matrices. The main researchcontents are as follows.(i) The two sample test of Chen&Qin[37] is extended to the test on the problemof k-sample Behrens-Fisher. The asymptotic null and non-null distributions of the newtests are derived. Theory and simulation results show that the new tests are morepowerful than existing tests in many cases.(ii) According to the idea of Chen&Qin[37], two new test statistics are proposedon the problem of k-sample Behrens-Fisher. The limit null and non-null distributionsare obtained. The new tests can be viewed as extensions of Chen&Qin’s test[37]. Bysimulation comparisons, it is demonstrated that the new tests are more competitiveones.(iii) For testing linear hypotheses on mean vectors of normal populations with un-equal covariance matrices, three new test statistics are proposed by Bennett’s transform-ation[68] in view of Bai&Saranadasa[36], Chen&Qin[37], Srivastava et al.[38]-[39] andPark&Ayyala’s[40] work. The asymptotic null and non-null distributions are proved.As a special case, the new tests can be used to solve Behrens-Fisher problem, whichowns the same asymptotic sizes and powers as those of Chen&Qin[37] and Srivastavaet al.[43], respectively. We illustrate benefits of the proposed procedure via simulationcomparisons which show that the new tests are comparable to, and are more powerfulthan the existing tests in many cases.