Adjustments of the profile likelihood from a new perspective

Various modifications of the profile likelihood have been proposed over the past 20 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 show that various modifications of the profile likelihood arise by refining the estimation process. In particular, we show that bias reduction of the profile loglikelihood as an estimate of the expectation of the least favourable target loglikelihood gives adjustments that agree to second order with the adjustment proposed by Severini [1998a. Biometrika 85, 403–411] as an easily computable approximation to the modified profile loglikelihood.