Statistical applications of random matrix theory: comparison of two populations I

This paper investigates a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent 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 random matrix theory, we concentrate on the largest eigenvalues of the matrices in order to determine significance. To avoid false rejections we must guard against residual spikes and need a sufficiently precise description of the behaviour of the largest eigenvalues under the null hypothesis. In this paper, we lay a foundation by treating alternatives based on perturbations of order $1$, that is, a single large eigenvalue. Our statistic allows the user to test the equality of two populations. Future work will extend the result to perturbations of order $k$ and demonstrate conservativeness of the procedure for more general matrices.

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