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This paper investigates a statistical procedure for testing the equality oftwo independently estimated covariance matrices when the number of potentiallydependent data vectors is large and proportional to the size of the vectors,that is, the number of variables. Inspired by the spike models used in randommatrix theory, we concentrate on the largest eigenvalues of the matrices inorder to determine significant differences. To avoid false rejections we mustguard against residual spikes and need a sufficiently precise description ofthe properties of the largest eigenvalues under the null hypothesis. In thispaper, we extend arXiv:2002.12741 for perturbation of order $1$ andarXiv:2002.12703 studying simpler statistic. The residual spike introduce inthe first paper is investigated and leads to a statistic that results in a goodtest of equality of two populations. Simulations show that this new test doesnot rely on some hypotheses that were necessary for the proofs and in thesecond paper.

Statistical applications of Random matrix theory: comparison of two populations III