A comparison of the accuracy of saddlepoint conditional cumulative distribution function approximations

TL;DR

This paper compares various saddlepoint approximation methods for the conditional distribution of the likelihood ratio statistic, evaluating their accuracy in hypothesis testing for exponential mean ratios.

Contribution

It systematically assesses multiple saddlepoint approximation techniques, including Barndorff-Nielsen, Severini, and DiCiccio and Martin modifications, using both Barndorff-Nielsen and Lugannani-Rice formats.

Findings

Barndorff-Nielsen's modified statistic shows improved accuracy.

Lugannani-Rice format generally outperforms Barndorff-Nielsen format.

Approximation accuracy varies with the method and format used.

Abstract

Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic R. The standard normal approximation to the conditional distribution of R typically has error of order O(n^{-1/2}), where n is the sample size. There are several modifications for R, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations.