On the Existence of Moments of Ratios of Quadratic Forms

We obtain simple and generally applicable conditions for the existence of mixed moments E ([ X′ AX ]″/[ X′ BX ]U ) of the ratio of quadratic forms T = X ′ AX /X ′ BX where A and B are n × n symmetric matrices and X is a random n -vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.