Exponential Linear Models: A Two-Pass Procedure for Saddlepoint Approximation

Abstract

SUMMARY For an exponential linear model, the saddlepoint method gives accurate approximations for the density of the minimal sufficient statistic or maximum likelihood estimate, and for the corresponding distribution functions. In this paper we describe a simple numerical procedure that constructs such approximations for a real parameter in an exponential linear model, using only a two-pass calculation on the observed likelihood function for the original data. Simple examples of the numerical procedure are discussed, but we take the general accuracy of the saddlepoint procedure as given. An immediate application of this is to exponential family models, where inference for a component of the canonical parameter is to be based on the conditional density of the corresponding component of the sufficient statistic, given the remaining components. This conditional density is also of exponential family form, but its functional form and cumulant-generating function may not be accessible. The procedure is applied to the corresponding likelihood, approximated as the full likelihood divided by an approximate marginal likelihood obtained from Barndorff-Nielsen's formula. A double saddlepoint approximation provides another means of bypassing this difficulty. The computational procedure is also examined as a numerical procedure for obtaining the saddlepoint approximation to the Fourier inversion of a characteristic function. As such it is a two-pass calculation on a table of the cumulant-generating function.