On interval and point estimators based on a penalization of the modified profile likelihood

Various modifications of the profile likelihood have been proposed in the literature. Despite modified profile likelihood methods have better properties than those based on the profile likelihood, the signed likelihood ratio statistic based on the modified profile likelihood has a standard normal distribution only to first order, and it can be inaccurate in particular in models with many nuisance parameters. In this paper we propose an adjustment of the profile likelihood from a new perspective. The idea is to resort to suitable default priors on the parameter of interest only to be used as non-negative weight functions in order to modify the modified profile likelihood. In particular, we focus on matching priors, i.e. priors on the parameter of interest only for which there is an agreement between frequentist and Bayesian inference, derived from modified profile likelihoods. The proposed modified profile likelihood has desiderable inferential properties: the corresponding signed likelihood ratio statistic is standard normal to second order and the correponding maximizer is a refinement of the maximum likelihood estimator, which improves its small sample properties. Examples illustrate the proposed modified profile likelihood and outline its improvement over its counterparts.