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

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