The Annals of Statistics

A Bernstein–von Mises theorem in the nonparametric right-censoring model

Yongdai Kim and Jaeyong Lee

Full-text: Open access


In the recent Bayesian nonparametric literature, many examples have been reported in which Bayesian estimators and posterior distributions do not achieve the optimal convergence rate, indicating that the Bernstein–von Mises theorem does not hold. In this article, we give a positive result in this direction by showing that the Bernstein–von Mises theorem holds in survival models for a large class of prior processes neutral to the right. We also show that, for an arbitrarily given convergence rate n−α with 0<α≤1/2, a prior process neutral to the right can be chosen so that its posterior distribution achieves the convergence rate n−α.

Article information

Ann. Statist., Volume 32, Number 4 (2004), 1492-1512.

First available in Project Euclid: 4 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62G20: Asymptotic properties 62N01: Censored data models

Bernstein–von Mises theorem neutral to the right process survival model


Kim, Yongdai; Lee, Jaeyong. A Bernstein–von Mises theorem in the nonparametric right-censoring model. Ann. Statist. 32 (2004), no. 4, 1492--1512. doi:10.1214/009053604000000526.

Export citation


  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536--561.
  • Breiman, L. (1968). Probability. Addison--Wesley, Reading, MA.
  • Brunner, L. J. and Lo, A. Y. (1996). Limiting posterior distributions under mixture of conjugate priors. Statist. Sinica 6 187--197.
  • Conti, P. L. (1999). Large sample Bayesian analysis for $\mathrmGeo/\mathrmG/1$ discrete-time queueing models. Ann. Statist. 27 1785--1807.
  • Cox, D. D. (1993). An analysis of Bayesian inference for nonparametric regression. Ann. Statist. 21 903--923.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Diaconis, P. and Freedman, D. A. (1986). On the consistency of Bayes estimates (with discussion). Ann. Statist. 14 1--67.
  • Diaconis, P. and Freedman, D. A. (1998). Consistency of Bayes estimates for nonparametric regression: Normal theory. Bernoulli 4 411--444.
  • Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183--201.
  • Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Ann. Statist. 7 163--186.
  • Freedman, D. A. (1963). On the asymptotic behavior of Bayes estimates in the discrete case. Ann. Math. Statist. 34 1386--1403.
  • Freedman, D. A. (1999). On the Bernstein--von Mises theorem with infinite-dimensional parameters. Ann. Statist. 27 1119--1140.
  • Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston.
  • Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143--158.
  • Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500--531.
  • Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I (with discussion). Scand. J. Statist. 16 97--128.
  • Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259--1294.
  • Jacod, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, New York.
  • Kalbfleisch, J. D. (1978). Nonparametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214--221.
  • Kim, Y. (1999). Nonparametric Bayesian estimators for counting processes. Ann. Statist. 27 562--588.
  • Kim, Y. and Lee, J. (2001). On posterior consistency of survival models. Ann. Statist. 29 666--686.
  • Lee, J. and Kim, Y. (2004). A new algorithm to generate beta processes. Comput. Statist. Data Anal. To appear.
  • Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55--66.
  • Lo, A. Y. (1983). Weak convergence for Dirichlet processes. Sankhyā Ser. A 45 105--111.
  • Lo, A. Y. (1986). A remark on the limiting posterior distribution of the multiparameter Dirichlet process. Sankhyā Ser. A 48 247--249.
  • Lo, A. Y. (1993). A Bayesian bootstrap for censored data. Ann. Statist. 21 100--123.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Schervish, M. J. (1995). Theory of Statistics. Springer, New York.
  • Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10--26.
  • Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687--714.
  • Tsiatis, A. A. (1981). A large sample study of Cox's regression model. Ann. Statist. 9 93--108.
  • Zhao, L. H. (2000). Bayesian aspects of some nonparametric problems. Ann. Statist. 28 532--551.