Approximations of marginal tail probabilities and inference for scalar parameters

Abstract

In many situations, inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density. For such inference, accurate approximations of marginal tail probabilities are useful to avoid high-dimensional integrals. Two tail probability approximations are developed in this paper. Numerical results given for conditional inference in location-scale and linear regression models show the approximations to be generally accurate even for small sample sizes.