Higher moments of matrix quadratic form

Abstract

880-01The importance of multivariate analysis in statistics has been recognized owing to its wide range of applications. And the matrix quadratic form plays a central role in multivariate analysis. The matrix quadratic form frequently appears in multivariate regression analysis in statistical inferrence, multivariate analysis of variance and test of statistical hypotheses, and has been much studied since 1980``s. The moment problem of matrix quadratic form has attracted the attentions of many researchers. The theory of matrix quadratic form, however, leaves much to be desired because, in general, the enormous size of the matrix to be considered often complicates computations, and in some cases makes it impossible to compute. When a random matrix $X$ is distributed the multivariate normal, $XAX``$ is called the matrix quadratic form, where $A$ is a constant matrix. In 1987, Neudecker and Wansbeek found the second moment of matrix quadratic form. In 1988 von Rosen solved the problem of any higher moment of the random matrix $X$, and as an application found the second moment of the matrix quadratic form by a method different from that of Neudecker and Wansbeek (1987). Using the result of von Rosen (1988), Tracy and Sultan found the third moment of matrix quadratic form in 1993. In this paper we derive some properties of the Kronecker product, vec operator and commutation matrix which are important tools in the moments of matrix quadratic form, and find any higher moment of matrix quadratic form by applying these properties and the result of von Rosen (1988). Morever, as an application of the general form of the higher moment of the matrix quadratic form we calculate any higher moment of the non-central Wishart distribution. Also, we provide various examples illustrating ours results. The results of this paper reveal novel formulae of matrix algebra and may shed light on the unsolvable problems.