Test for diagonal symmetry in high dimension

Abstract

Utilizing the energy distance and energy statistics, Sang and Dang (2020) proposed a test statistic as a difference of two U-statistics for the diagonal symmetry test of a p-vector X. Under the regular setting where the dimensionality of the random vector is fixed, the test statistic is a degenerate U-statistic and hence converges to a mixture of chi-squared distributions. In this paper, we test the diagonal symmetry of X in a more realistic setting where both the sample size and the dimensionality are diverging to infinity. Our theoretical results reveal that the degenerate U-statistic admits a central limit theorem in the high dimensional setting and the accuracy of normal approximation can increase with dimensionality. We then construct a powerful and consistent test for the diagonal symmetry problem based on the asymptotic normality. Simulation studies are conducted to illustrate the performances of the test.