A two sample test in high dimensional data

In this paper we propose a test for testing the equality of the mean vectors of two groups with unequal covariance matrices based on N1N1 and N2N2 independently distributed pp-dimensional observation vectors. It will be assumed that N1N1 observation vectors from the first group are normally distributed with mean vector μ1 and covariance matrix Σ1. Similarly, the N2N2 observation vectors from the second group are normally distributed with mean vector μ2 and covariance matrix Σ2. We propose a test for testing the hypothesis that μ1=μ2. This test is invariant under the group of p×pp×p nonsingular diagonal matrices. The asymptotic distribution is obtained as (N1,N2,p)→∞(N1,N2,p)→∞ and N1/(N1+N2)→k∈(0,1)N1/(N1+N2)→k∈(0,1) but N1/pN1/p and N2/pN2/p may go to zero or infinity. It is compared with a recently proposed non-invariant test. It is shown that the proposed test performs the best.