On testing for an identity covariance matrix when the dimensionality equals or exceeds the sample size

This article explores the problem of testing the hypothesis that the covariance matrix is an identity matrix when the dimensionality is equal to the sample size or larger. Two new test statistics are proposed under comparable assumptions to those statistics in the literature. The asymptotic distribution of the proposed test statistics are found and are shown to be consistent in the general asymptotic framework. An extensive simulation study shows the newly proposed tests are comparable to, and in some cases more powerful than, the tests for an identity covariance matrix currently in the literature.

► The problem of testing for an identity covariance matrix is discussed.
► An estimator for the third arithmetic mean of the eigenvalues of the covariance matrix is derived.
► Estimators for the first four arithmetic means are found to be asymptotically jointly normal.
► Two statistics are developed for testing the identity hypothesis.
► A simulation study shows the statistics are effective.