  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.
