A nonparametric test for block-diagonal covariance structure in high dimension and small samples

In this paper we consider the problem of hypothesis testing for block-diagonal structure of high-dimensional covariance matrix. We develop a bias correction to the existing scalar transform invariant test statistic that is constructed based on an empirical distance between the full and a block diagonal covariance matrix, without requiring any specific parametric distribution such as the normality assumption. Under the high-dimensional null hypothesis and the scenario of the alternatives, which allows power evaluations, we derive the asymptotic distribution of the proposed test statistic without specifying an explicit relationship between the dimension and the sample size. Monte Carlo simulation studies demonstrate that it has good size and power in a wide range of settings. A real data example is also considered to illustrate the efficacy of the approach.