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 N 1 and N 2 independently distributed p -dimensional observation vectors. It will be assumed that N 1 observation vectors from the first group are normally distributed with mean vector μ 1 and covariance matrix Σ 1 . Similarly, the N 2 observation vectors from the second group are normally distributed with mean vector μ 2 and covariance matrix Σ 2 . We propose a test for testing the hypothesis that μ 1 = μ 2 . This test is invariant under the group of p × p nonsingular diagonal matrices. The asymptotic distribution is obtained as ( N 1 , N 2 , p ) → ∞ and N 1 / ( N 1 + N 2 ) → k ∈ ( 0 , 1 ) but N 1 / p and N 2 / 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.