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.


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


Zbl 1142.62030
Full Text: DOI arXiv


[1] Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. Roy. Soc. London Ser. A 160 268-282. · Zbl 0016.41201
[2] Birman, M. S. and Solomjak, M. J. (1967). Piece-wise polynomial approximations of functions of the classes W p \alpha . Mat. Sbornik 3 295-317.
[3] Cheng, G. (2006). Higher order semiparametric frequentist inference and the profile sampler. Ph.D. thesis, Dept. Statistics, Univ. Wisconsin-Madison.
[4] Cheng, G. and Kosorok, M. R. (2008). Higher order semiparametric frequentist inference with the profile sampler. Ann. Statist. 36 1786-1818. · Zbl 1142.62030
[5] Cox, D. R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 187-220. JSTOR: · Zbl 0243.62041
[6] Dalalyan, A. S., Golubev, G. K. and Tsybakov, A. B. (2006). Penalized maximum likelihood and semiparametric second-order efficiency. Ann. Statist. 34 169-201. · Zbl 1091.62020
[7] Groeneboom, P. (1991). Nonparametric maximum likelihood estimators for interval censoring and deconvolution. Technical Report 378, Dept. Statistics, Stanford Univ.
[8] Härdle, W. and Tsybakov, A. B. (1993). How sensitive are average derivatives? J. Econometrics 58 31-48. · Zbl 0772.62021
[9] Huang, J. (1996). Efficient estimation for the Cox model with interval censoring. Ann. Statist. 24 540-568. · Zbl 0859.62032
[10] Kass, R. E. and Wasserman, L. (1996). Formal rules for selecting prior distributions: A review and annotated bibliography. J. Amer. Statist. Assoc. 91 1343-1370. · Zbl 0884.62007
[11] Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference . Springer, New York. · Zbl 1180.62137
[12] Kosorok, M. R., Lee, B. L. and Fine, J. P. (2004). Robust inference for univariate proportional hazards frailty regression models. Ann. Statist. 32 1448-1491. · Zbl 1047.62090
[13] Kuo, H. H. (1975). Gaussian Measure on Banach Spaces . Springer, Berlin. · Zbl 0306.28010
[14] Lee, B. L., Kosorok, M. R. and Fine, J. P. (2005). The profile sampler. J. Amer. Statist. Assoc. 100 960-969. · Zbl 1117.62380
[15] Ma, S. and Kosorok, M. R. (2005). Penalized log-likelihood estimation for partly linear transformation models with current status data. Ann. Statist. 33 2256-2290. · Zbl 1086.62056
[16] Murphy, S. A. and Van der Vaart, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25 1471-1509. · Zbl 0928.62036
[17] Murphy, S. A. and Van der Vaart, A. W. (1999). Observed information in semiparametric models. Bernoulli 5 381-412. · Zbl 0954.62036
[18] Murphy, S. A. and Van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 1461-1474. JSTOR: · Zbl 0995.62033
[19] Mukerjee, R. and Ghosh, M. (1997). Second-order probability matching priors. Biometrika 84 970-975. JSTOR: · Zbl 0895.62003
[20] Reid, N. (1995). Likelihood and Bayesian approximation methods (with discussion). In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 351-368. Oxford Univ. Press.
[21] Shen, X. (2002). Asymptotic normality of semiparametric and nonparametric posterior distributions. J. Amer. Statist. Assoc. 97 222-235. JSTOR: · Zbl 1073.62517
[22] van de Geer, S. (2000). Empirical Processes in M-estimation . Cambridge Univ. Press. · Zbl 0953.62049
[23] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes : With Applications to Statistics . Springer, New York. · Zbl 0862.60002
[24] Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihood. J. Roy. Statist. Soc. Ser. B 25 318-329. JSTOR: · Zbl 0117.14205
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