On high-dimensional two sample mean testing statistics: a comparative study with a data adaptive choice of coefficient vector

The key issues involved in two sample tests in high dimensional problems arise due to large dimension of the mean vector for a relatively small sample size. Recently, Wang et al. (Stat Sin 23:667–690, 2013) proposed a jackknife empirical likelihood test that works under weak assumptions on the dimension of variables (p), and showed that the test statistic has a chi-square limit regardless of whether p is finite or diverges. The sufficient condition required for this statistic is still restrictive. In this paper we significantly relax the sufficient condition for the asymptotic chi-square limit with models allowing flexible dependence structures and derive simpler alternative statistics for testing the equality of two high dimensional means. The proposed statistics have a chi-squared distribution or the maximum of two independent chi-square statistics as their limiting distributions, and the asymptotic results hold for either finite or divergent p. We also propose a data-adaptive method to select the coefficient vector, and compare the various methods in simulation studies. The proposed choice of coefficient vector substantially increases power in the simulation.