Simple Modifications for Signed Roots of Likelihood Ratio Statistics

Abstract

SUMMARY In the context of inference about a scalar parameter in the presence of nuisance parameters, some simple modifications for the signed root of the log-likelihood ratio statistic R are developed that reduce the order of error in the standard normal approximation to the distribution of R from O(n -1/2) to O(n -1). Barndorff-Nielsen has introduced a variable U such that the error in the standard normal approximation to the distribution of R + R -1 log(U/R) is of order O(n -3/2), but calculation of U requires the specification of an exact or approximate ancillary statistic A. This paper proposes an alternative variable to U, denoted by T, that is available without knowledge of A and satisfies T = U + Op(n -1) in general. Thus the standard normal approximation to the distribution of R + R -1 log(T/R) has error of order O(n -1), and it can be used to construct approximate confidence limits having coverage error of order O(n -1). In certain cases, however, T and U are identical. The derivation of T involves the Bayesian approach to constructing confidence limits considered by Welch and Peers, and Peers. Similar modifications for the signed root of the conditional likelihood ratio statistic are also developed, and these modifications are seen to be useful when a large number of nuisance parameters are present. Several examples are presented, including inference for natural parameters in exponential models and inference about location-scale models with type II censoring. In each case, the relationship between T and U is discussed. Numerical examples are also given, including inference for regression models, inference about the means of log-normal distributions and inference for exponential lifetime models with type I censoring, where Barndorff-Nielsen's variable U is not available.