The Annals of Statistics

Nonparametric Bayesian estimators for counting processes

Yongdai Kim

Full-text: Open access


This paper is concerned with nonparametric Bayesian inference of the Aalen’s multiplicative counting process model. For a desired nonparametric prior distribution of the cumulative intensity function, a class of Lévy processes is considered, and it is shown that the class of Lévy processes is conjugate for the multiplicative counting process model, and formulas for obtaining a posterior process are derived. Finally, our results are applied to several practically important models such as one point processes for right-censored data, Poisson processes and Markov processes.

Article information

Ann. Statist., Volume 27, Number 2 (1999), 562-588.

First available in Project Euclid: 5 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 60G55: Point processes

Nonparametric Bayesian estimator multiplicative counting process Lévy process.


Kim, Yongdai. Nonparametric Bayesian estimators for counting processes. Ann. Statist. 27 (1999), no. 2, 562--588. doi:10.1214/aos/1018031207.

Export citation


  • Aalen, O. O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701-726.
  • Anderson, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201.
  • Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Belmont, CA.
  • 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 probl emes de martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, New York.
  • Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • Kao, E. and Smith, M. S. (1993). On renewal processes relating to counter models: the case of phase-type interarrival times. J. Appl. Probab. 30 175-183.
  • Karr, A. F. (1986). Point Processes and Their Statistical Inference. Dekker, New York.
  • Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point process. Z. Wahrsch. Verw. Gebiete 59 55-66.
  • Lo, A. Y. (1992). Bayesian inference for Poisson process models with censored data. J. Nonparametr. Statist. 2 71-80.
  • Pyke, R. (1958). On renewal processes related to type I and type II counter models. Ann. Math. Statist. 29 737-754.
  • Shiryaev, A. N. (1991). Probability. Springer, New York.
  • Sweeting, T. J. (1989). On conditional weak convergence. J. Theoret. Probab. 2 461-474.