  Rank estimation is a classical model order selection problem that arises in a
variety of important statistical signal and array processing systems, yet is
addressed relatively infrequently in the extant literature. Here we present
sample covariance asymptotics stemming from random matrix theory, and bring
them to bear on the problem of optimal rank estimation in the context of the
standard array observation model with additive white Gaussian noise. The most
significant of these results demonstrates the existence of a phase transition
threshold, below which eigenvalues and associated eigenvectors of the sample
covariance fail to provide any information on population eigenvalues. We then
develop a decision-theoretic rank estimation framework that leads to a simple
ordered selection rule based on thresholding; in contrast to competing
approaches, however, it admits asymptotic minimax optimality and is free of
tuning parameters. We analyze the asymptotic performance of our rank selection
procedure and conclude with a brief simulation study demonstrating its
practical efficacy in the context of subspace tracking.
