Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality

Ahmad and von Rosen (2014 Ahmad, M. R. (2014). A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting. Annals of the Institute of Statistical Mathematics 66:33–61.Crossref], Web of Science ®] , Google Scholar]) presented test statistics for sphericity and identity of the covariance matrix of a multivariate normal distribution when the dimension, p, exceeds the sample size, n. In this note, we show that their statistics are robust to normality assumption, when normality is replaced with certain mild assumptions on the traces of the covariance matrix. Under such assumptions, the test statistics are shown to follow the same asymptotic normal distribution as under normality for large p, also when p > >n. The asymptotic normality is proved using the theory of U-statistics, and is based on very general conditions, particularly avoiding any relationship between n and p.