  In this paper, we study the problem of testing the mean vectors of high
dimensional data in both one-sample and two-sample cases. The proposed testing
procedures employ maximum-type statistics and the parametric bootstrap
techniques to compute the critical values. Different from the existing tests
that heavily rely on the structural conditions on the unknown covariance
matrices, the proposed tests allow general covariance structures of the data
and therefore enjoy wide scope of applicability in practice. To enhance powers
of the tests against sparse alternatives, we further propose two-step
procedures with a preliminary feature screening step. Theoretical properties of
the proposed tests are investigated. Through extensive numerical experiments on
synthetic datasets and an human acute lymphoblastic leukemia gene expression
dataset, we illustrate the performance of the new tests and how they may
provide assistance on detecting disease-associated gene-sets. The proposed
methods have been implemented in an R-package HDtest and are available on CRAN.
