Asymptotic Expansions for Distributions of the Roots of Two Matrices from Classical and Complex Gaussian Populations

Abstract

The distribution of the characteristic (ch.) roots of a sample covariance matrix $\mathbf{S}$ (one-sample case) or the matrix $\mathbf{S}_1\mathbf{S}_2^{-1}$ (two-sample case, see Section 3) depends on a definite integral over the group of orthogonal (in the complex case replaced by unitary) matrices. This integral, either in the one-sample case or the two-sample case, involves the ch. roots of both the population and sample matrices. Usually the integral in either case is expressed as a hypergeometric series involving zonal polynomials [4], [6]. Unfortunately, these series converge slowly unless the ch. roots of the argument matrices lie in very limited ranges. Furthermore, the computations of these series are not so easy and not convenient for further development. In the one-sample real case, Anderson [1] has obtained an asymptotic expansion for the distribution of the ch. roots of the sample covariance matrix. In the two-sample case, however, the situation is more complicated. Chang [2] has obtained an asymptotic expansion for the first term. In Section 3 and Section 4 of this paper, we extend Chang's results obtaining the second term and also derive a general formula which includes the formulae of Anderson [1], James [7], Chang [2] and Roy [12] as limiting or special cases. In Section 5 we are dealing with the asymptotic expansions in the two-sample case in the complex Gaussian population, from which the one-sample results are obtained as limiting cases. Finally, Section 6 gives a comparison of the four asymptotic expansions.