Statistical eigen-inference from large Wishart matrices

The asymptotic behavior of the eigenvalues of a sample covariance matrix is described when the observations are from a zero mean multivariate (real or complex) normal distribution whose covariance matrix has population eigenvalues of arbitrary multiplicity. In particular, the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix from the limiting quantities is shown. Concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations are presented. Tests of hypotheses for the population eigenvalues are developed and a technique for inferring the population eigenvalues (without requiring any assumptions on the population eigenvectors) is proposed that exploits this asymptotic normality of the trace of powers of the sample covariance matrix. Numerical simulations demonstrate the robustness of the proposed techniques in techniques in high-dimensional, (relatively) small sample size settings and the superiority over alternate procedures found in the literature for the special cases where a direct comparison can be made. The improved performance is a consequence of the fact that the proposed inference procedures are “global” (in a sense that we describe) and exploit “global” information thereby overcoming the inherent biases that cripple classical “local” inference procedures which rely on “local” information.