Higher order semiparametric frequentist inference with the profile sampler

We consider higher order frequentist inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution. The first order validity of this procedure established by Lee, Kosorok and Fine in [J. American Statist. Assoc. 100 (2005) 960--969] is extended to second-order validity in the setting where the infinite-dimensional nuisance parameter achieves the parametric rate. Specifically, we obtain higher order estimates of the maximum profile likelihood estimator and of the efficient Fisher information. Moreover, we prove that an exact frequentist confidence interval for the parametric component at level $\alpha$ can be estimated by the $\alpha$-level credible set from the profile sampler with an error of order $O_P(n^{-1})$. Simulation studies are used to assess second-order asymptotic validity of the profile sampler. As far as we are aware, these are the first higher order accuracy results for semiparametric frequentist inference.