Covariance matrix testing in high dimension using random projections

TL;DR

This paper introduces CRAMP, a new high-dimensional covariance matrix test using random projections, which improves computational efficiency and error control compared to existing methods, demonstrated through simulations and gene data applications.

Contribution

The paper proposes CRAMP, a novel random projection-based test for high-dimensional covariance matrices, addressing computational and error control issues in existing methods.

Findings

CRAMP outperforms existing high-dimensional tests in simulations.

CRAMP effectively controls type I error in high dimensions.

Application to gene expression data demonstrates practical utility.

Abstract

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the covariance matrix in high dimensions is a challenging problem as the traditional multivariate asymptotic theory is no longer valid.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.