A second order adjustment to the profile likelihood in the case of a multi-dimensional parameter of interest

Research Report SRR95-038

A second order adjustment to the profile likelihood in the case of a multi-dimensional parameter of interest

Steven E. Stern

Abstract: Inference in the presence of nuisance parameters is often carried out using the chi-squared approximation to the profile likelihood ratio statistic. However, in small samples, the accuracy of such procedures may be poor, due in part to the fact that the profile likelihood does not behave as a true likelihood, in particular having a profile score bias and information bias which do not vanish. To better account for the presence of nuisance parameters, various authors have suggested basing inference on an additively adjusted version of the profile likelihood function. Each of these adjustments to the profile likelihood generally has the effect of reducing the bias of the associated profile score statistic. However, these adjustments are not applicable outside of the specific parametric framework for which they were developed. In particular, it is often difficult or even impossible to apply them in the case where the parameter about which inference is desired is multi-dimensional. In this paper, we propose a new adjustment function which leads to an adjusted profile likelihood having reduced score and information biases and is readily applicable to a general parametric framework, including the case of vector-valued parameters of interest. Examples are given to examine the performance of the new adjusted profile likelihood in small samples, and also to compare its performance with other adjusted profile likelihoods.

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