On the moments of ratios of quadratic forms in normal random variables

In this paper, we present both integral and infinite series expressions of μ q p ≡ E [ ( x ′ A x ) p / ( x ′ B x ) q ] when x ∼ N ( μ , I n ) , where p , q are nonnegative real numbers, A is a symmetric matrix, and B is a positive semi-definite matrix. We also present efficient numerical methods for computing μ q p under each approach.