Comparing approximations to the expectation of a ratio of quadratic forms in normal variables

Approximations to the expectation of a ratio of quadratic forms in normal variables have been proposed by Ullah and Srivastava (1994) and Lieberman (1994), the former suggests a small disturbance approximation and the latter a Laplace approximation. Another approximation based on Nagars (1959) method is derived. Each proposal relies upon taking the expectation of an approximation to the ratio. This paper considers the expectation of the particular ratio (Formula presented.) where x is an n-dimensional normal random vector with non-zero mean and A is a symmetric matrix. The exact formula for E(r) is given in Smith (1993) and involves confluent hypergeometric functions. Asymptotic expansions for these functions are applied enabling a large n approximation and a small disturbance approximation to E(r) to be developed. The accuracies of the approximations are studied in a variety of settings. It is demonstrated that approximations based on large n asymptotics, such as the Nagar, are more reliable than other methods.