Saddlepoint Approximation for Multivariate Cumulative Distribution Functions and Probability Computations in Sampling Theory and Outlier Testing

Abstract

Four multivariate distributions commonly arise in sampling theory: the multinomial, multivariate hypergeometric, Dirichlet, and multivariate Pólya distributions. Second-order saddlepoint approximations are given for approximating these multivariate cumulative distribution functions (cdf's) in their most general settings. Probabilities of rectangular regions associated with these cdf's are also approximated directly using second-order saddlepoint methods. All the approximations follow from characterizations of the multivariate distributions as conditional distributions. Applications to outlier discordancy tests and slippage tests are discussed.