The Profile Sampler

Abstract

We consider frequentist inference for the parametric component θ separately from the nuisance parameter η in semiparametric models based on sampling from the posterior of the profile likelihood. We prove that this procedure gives a first-order–correct approximation to the maximum likelihood estimator and consistent estimation of the efficient Fisher information for θ, without computing derivatives or using complicated numerical approximations. An exact Bayesian interpretation is established under a certain data-dependent prior. The sampler is useful in particular when the nuisance parameter is not estimable at the rate, where neither bootstrap validity nor general automatic variance estimation has been theoretically justified. Even when the nuisance parameter is consistent and the bootstrap is known to be valid, the proposed Markov chain Monte Carlo procedure can yield computational savings, because maximization of the likelihood is not required. The theory is verified for three examples. The methods are shown to perform well in simulations, and their practical utility is illustrated in two data analyses.