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General frequentist properties of the posterior profile distribution. (English) Zbl 1142.62031

Summary: In this paper, inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution is thoroughly investigated from the frequentist viewpoint. The higher-order validity of the profile sampler obtained by G. Cheng and M. R. Kosorok [Ann. Stat. 36, No. 4, 1786–1818 (2008; Zbl 1142.62030)] is extended to semiparametric models in which the infinite dimensional nuisance parameter may not have a root-\(n\) convergence rate. This is a nontrivial extension because it requires 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. Simulation studies are used to verify this theoretical result. We also establish that an exact frequentist confidence interval obtained by inverting the profile log-likelihood ratio can be estimated with higher-order accuracy by the credible set of the same type obtained from the posterior profile distribution. Our theory is verified for several specific examples.

MSC:

62G20 Asymptotic properties of nonparametric inference
62F25 Parametric tolerance and confidence regions
62F12 Asymptotic properties of parametric estimators
62A01 Foundations and philosophical topics in statistics

Citations:

Zbl 1142.62030
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