Improved profile likelihood inference

Inference about a multiparameter of interest in the presence of nuisance parameters is oftentimes based on the profile likelihood function. However, it does not behave as a true likelihood function and several adjustments to the profile likelihood function have been proposed. In this paper, we consider an additive adjustment that aims at reducing the score and information biases of the score function from order O(1) to order O(n-1). The adjustment is applicable in wide generality since it allows both the interest and nuisance parameters to be vector-valued. We derive a Bartlett correction for the adjusted profile likelihood ratio statistic and obtain closed-form expressions for the class of the generalized linear models. A simulation study compares the performance of the usual profile likelihood ratio test, the adjusted profile likelihood ratio test and the associated Bartlett-corrected test.