Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.