High-dimensional asymptotic expansion of the null distribution for Schott’s test statistic for complete independence of normal random variables

This article is concerned with the testing complete independence for the elements of observed vector. Schott proposed the testing statistic T and gave limiting null distribution under the high-dimensional asymptotic framework that the sample size n and the dimensionality p go to infinity together while p/n converges to a positive constant. In this article we give a one-term asymptotic expansion of the null distribution for T as $\min \{n,p\}$ tends toward infinity. We derive a correction of the critical point for Schott’s test based on this expansion. The finite sample size and dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Schott’s test.