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 (2005) 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 e 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 OP(n −1 ). As far as we are aware, these results are the first higher order frequentist results obtained for semiparametric estimation. A fully Bayesian interpretation is established under a certain data dependent prior. The theory is verified for three specific examples. 1. Introduction. The focus of this paper is on higher order frequentist inference for the parametric component θ of a semiparametric model. In addition to the d-dimensional Euclidean parameter θ, semiparametric models also have an infinite-dimensional parameter η which is sometimes called the “nuisance” parameter. A classic example is the Cox proportional hazards model for right-censored survival data [7], where interest focuses on the log hazard ratios θ for the regression covariate vector z. The integrated baseline hazard function η is the infinite-dimensional nuisance parameter. The involvement of an infinite-dimensional nuisance parameter in semiparametric models generally complicates maximum likelihood inference for θ. In particular, estimating the limiting variance of √ n( ˆ θn − θ0), where θ0 is the true value of θ, usually requires estimating an infinite-dimensional operator. Of course, this is not a problem with the Cox model since the profile likelihood for θ does not involve the nuisance parameter, but for most semiparametric models this simplification of the profile likelihood does not occur and ∗ Supported in part by Grant CA075142.