## The Annals of Statistics

### Beta-Stacy processes and a generalization of the Pólya-urn scheme

#### Abstract

A random cumulative distribution function (cdf) F on $[0, \infty)$ from a beta-Stacy process is defined. It is shown to be neutral to the right and a generalization of the Dirichlet process. The posterior distribution is also a beta-Stacy process given independent and identically distributed (iid) observations, possibly with right censoring, from F. A generalization of the Pólya-urn scheme is introduced which characterizes the discrete beta-Stacy process.

#### Article information

Source
Ann. Statist., Volume 25, Number 4 (1997), 1762-1780.

Dates
First available in Project Euclid: 9 September 2002

https://projecteuclid.org/euclid.aos/1031594741

Digital Object Identifier
doi:10.1214/aos/1031594741

Mathematical Reviews number (MathSciNet)
MR1463574

Zentralblatt MATH identifier
0928.62067

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 60G09: Exchangeability

#### Citation

Walker, Stephen; Muliere, Pietro. Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25 (1997), no. 4, 1762--1780. doi:10.1214/aos/1031594741. https://projecteuclid.org/euclid.aos/1031594741

#### References

• Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via P´oly a-urn schemes. Ann. Statist. 1 353-355.
• Connor, R. J. and Mosimann, J. E. (1969). Concepts of independence for proportions with a generalisation of the Dirichlet distribution. J. Amer. Statist. Assoc. 64 194-206.
• Damien, P., Laud, P. and Smith, A. F. M. (1995). Random variate generation approximating infinitely divisible distributions with application to Bayesian inference. J. Roy. Statist. Soc. Ser. B 57 547-564.
• Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201.
• Dy kstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356-367.
• Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
• Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615-629.
• Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Ann. Statist. 7 163-186.
• Gill, R. D. and Johansen, S. (1990). A survey of product integration with a view toward application in survival analysis. Ann. Statist. 18 1501-1555.
• Hjort, N. L. (1990). Nonparametric Bay es estimators based on beta processes in models for life history data. Ann. Statist. 18 1259-1294.
• Lavine, M. (1992). Some aspects of P´oly a tree distributions for statistical modelling. Ann. Statist. 20 1222-1235.
• Lavine, M. (1994). More aspects of P´oly a trees for statistical modelling. Ann. Statist. 22 1161- 1176.
• L´evy, P. (1936). Th´eorie de l'Addition des Variables Al´eatoire. Gauthiers-Villars, Paris.
• Lo, A. Y. (1988). A Bayesian bootstrap for a finite population. Ann. Statist. 16 1684-1695.
• Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). P´oly a trees and random distributions. Ann. Statist. 20 1203-1221.
• Mihram, G. A. and Hultquist, R. A. (1967). A bivariate warning-time/failure-time distribution. J. Amer. Statist. Assoc. 62 589-599.
• Muliere, P. and Walker, S. G. (1995). Extending the family of Bayesian bootstraps and exchangeable urn schemes. J. Roy. Statist. Soc. Ser. B. To appear.
• Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Roy. Statist. Soc. Ser. B 55 3-23.
• Susarla, V. and Van Ry zin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete data. J. Amer. Statist. Assoc. 71 897-902.