Adjustements of the profile likelihood from a new perspective.

Various modifications of the profile likelihood have been proposed over the past twenty years. Their main theoretical basis is higher-order approximation of some target likelihood, defined by a suitable model reduction via conditioning or marginalisation, where the reduced model is indexed only by the parameter of interest. However, an exact reduced target likelihood exists only for special classes of models. In this paper, a general target likelihood is defined through model restriction along the least favourable curve in the parameter space. The profile likelihood can be seen as a purely estimative counterpart of this least favourable target likelihood. We will show that various modifications of the profile likelihood arise by refining the estimation process. In particular, bias reduction of the profile loglikelihood as an estimate of the expectation of the least favourable target loglikelihood gives adjustments that coincide to second order and agree with the available adjustments.