Title: Higher Order Semiparametric Frequentist Inference Based on the
Profile Sampler

Abstract: In this talk, we have systematically constructed a higher
 order frequentist validation of semiparametric estimation procedures
 through easy-to-implement Bayesian MCMC methodology. Specifically
 speaking, inference for the parametric component of a semiparametric
 model based on sampling from the posterior profile distribution,
 called "the profile sampler", is thoroughly investigated from
 frequentist viewpoint. We first derive the second order asympotic
 frequentist properties of the profile sampler in terms of
 distributions, moments and confidence intervals. Further, by a
 delicate analysis of the entropy of the semiparametric models
 involved, we find that the accuracy of inferences based on the
 profile sampler improves as the convergence rate of the nuisance
 parameter increases. From the above analysis, we notice that the
 estimation accuracy of the profile sampler method is intrinsically
 determined by the semiparametric model specifications. Therefore it
 is somewhat natural to !  question how to control the degree of
 accuracy. In the last section, we address this by proposing the
 penalized profile sampler method, in which we profile the penalized
 likelihood rather than the full likelihood. Using the penalized
 profile sampler procedure, we can achieve the desired estimation
 accuracy for the parameter of interest by adjusting the size of the
 assigned smoothing parameter.

The theoretical validity of the above procedures is illustrated in
several popular semiparametric models arising from Survival Analysis,
Epidemiology and Econometrics. As far as we are aware, the above
results are the first higher order frequentist inferences obtained for
semiparametric estimation.