A 2ND-ORDER ADJUSTMENT TO THE PROFILE LIKELIHOOD IN THE CASE OF A MULTIDIMENSIONAL PARAMETER OF INTEREST

Inference in the presence of nuisance parameters is often carried out by using the chi(2)-approximation to the profile likelihood ratio stat istic. However, in small samples, the accuracy of such procedures may be poor, in part because the profile Likelihood does not behave as a t rue likelihood, in particular having a profile score bias and informat ion bias which do not vanish. To account better for nuisance parameter s, various researchers have suggested that inference be based on an ad ditively adjusted version of the profile likelihood function. Each of these adjustments to the profile likelihood generally has the effect o f reducing the bias of the associated profile score statistic. However , these adjustments are not applicable outside the specific parametric framework for which they were developed. In particular, it is often d ifficult or even impossible to apply them where the parameter about wh ich inference is desired is multidimensional. In this paper, we propos e a new adjustment function which leads to an adjusted profile likelih ood having reduced score and information biases and is readily applica ble to a general parametric framework, including the case of vector-va lued parameters of interest. Examples are given to examine the perform ance of the new adjusted profile likelihood in small samples, and also to compare its performance with other adjusted profile likelihoods.