Eigenvalue tests and distributions for small sample order determination for complex Wishart matrices

Abstract

This is a theoretical thesis. The goal is to determine how many signal sources exist in the medium when constrained to using only a few samples. The need to make decisions based on only a few samples is motivated by the slow sound propagation speed and the time urgency to make decisions. This research treats the problem from the point of view of classical hypothesis testing assuming complex multivariate Gaussian random variables. This is the small sample complex principal components analysis problem. The critical issue is the derivation of probability density functions of appropriate test statistics. The goal has been partially achieved. The probability density functions for several important distributions have been derived. In particular, these include the distribution for the set of eigenvalues satisfying the generalized eigenvalue problem of two complex Wishart matrices, the matrix complex Gaussian distribution, a joint distribution needed to derive the density for the sphericity test statistic, the density function for the ratio of averages of disjoint sums of sequential eigenvalues of a complex Wishart matrix, and several tests based on the ratio of an arbitrary eigenvalue to the maximum, minimum, average, or sum of all the eigenvalues for a special case of the complex Wishart matrix. This thesis includes a derivation completely in the context of complex variables of the density function of the complex Wishart distribution and the distribution of its eigenvalues. It also includes a few minor results regarding zonal polynomials of complex matrix argument. A comprehensive development of the tools of statistics of complex variables for engineers and physicists is provided. This includes a study of complex matrix derivatives, changes of complex variables, and properties of the characteristic function of a complex multivariate random variable. A derivation of the complex Hotelling's T$\sp2$ test statistic and distribution useful for tests on means is given. A tutorial on Kiefer and Wolfowitz' application of the Lebesgue-Radon-Nikodym theorem for the estimation approach is provided.