Covariance matrix testing in high dimension using random projections

Estimation and hypothesis tests for the covariance matrix in high dimensions is a challenging problem as the traditional multivariate asymptotic theory is no longer valid. When the dimension is larger than or increasing with the sample size, standard likelihood based tests for the covariance matrix have poor performance. Existing high dimensional tests are either computationally expensive or have very weak control of type I error. In this paper, we propose a test procedure, CRAMP, for testing hypotheses involving one or more covariance matrices using random projections. Projecting the high dimensional data randomly into lower dimensional subspaces alleviates of the curse of dimensionality, allowing for the use of traditional multivariate tests. An extensive simulation study is performed to compare CRAMP against asymptotics-based high dimensional test procedures. An application of the proposed method to two gene expression data sets is presented.