Beta-stacy processes and a generalization of the Pólya-Urn scheme. (English) Zbl 0928.62067

Walker, Stephen; Muliere, Pietro

Ann. Stat. 25, No. 4, 1762-1780 (1997).

Let \({\mathcal F}\) be the space of cumulative distribution functions (cdfs) on \([0,\infty)\). This paper considers placing a probability distribution on \({\mathcal F}\) by defining a stochastic process \(F\) on \(([0,\infty),{\mathcal A})\), where \({\mathcal A}\) is the Borel \(\sigma\)-field of subsets, such that \(F(0)= 0\) a.s., \(F\) is a.s. nondecreasing, a.s. right continuous and \(\lim_{t\to\infty} F(t)=1\) a.s. Thus, with probability 1, the sample paths of \(F\) are cdf’s.
The purpose of this paper is twofold: (1) to introduce a new stochastic process which generalizes the Dirichlet process, in that more flexible prior beliefs are able to be represented, and, unlike the Dirichlet process, is conjugate to right censored observations, and (2) to introduce a generalization of the Pólya-urn scheme in order to characterize the discrete time version of the process. The property of conjugacy to right censored observations is also a feature of the beta process; however, with the beta process the statistician is required to consider hazard rates and cumulative hazards when constructing the prior. The beta-Stacy process only requires considerations on the distribution of the observations. The process is shown to be neutral to the right.
The paper is restricted to considering the estimation of an unknown cdf on \([0,\infty)\), although it is trivially extended to include \((-\infty, \infty)\). Finally, for ease of notation, \(F\) is written to mean either the cdf or the corresponding probability measure. The organization of the paper is as follows.
In Section 2 the process is defined and its connections with other processes given. We also provide an interpretation for the parameters of the process in terms of the mean and variance of \(F\). Section 3 considers the construction and posterior distributions of the discrete time process, while Section 4 considers the process in continuous time. In Section 4.4 we present a numerical example illustrating the process in continuous time. In Section 5 a generalization of Pólya’s urn scheme is introduced which characterizes the discrete time process.

MSC:

62M09	Non-Markovian processes: estimation
62C10	Bayesian problems; characterization of Bayes procedures
60G09	Exchangeability for stochastic processes

Keywords:

Bayesian nonparametrics; generalized Dirichlet distribution; generalized Polya-urn scheme; Levy process; neutral to the right process; Dirichlet process

References:

[1]	Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Póly a-urn schemes. Ann. Statist. 1 353-355. · Zbl 0276.62010 · doi:10.1214/aos/1176342372
[2]	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. JSTOR: · Zbl 0179.24101 · doi:10.2307/2283728
[3]	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. JSTOR: · Zbl 0827.60009
[4]	Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201. · Zbl 0279.60097 · doi:10.1214/aop/1176996703
[5]	Dy kstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356-367. · Zbl 0469.62077 · doi:10.1214/aos/1176345401
[6]	Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037 · doi:10.1214/aos/1176342360
[7]	Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615-629. · Zbl 0286.62008 · doi:10.1214/aos/1176342752
[8]	Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Ann. Statist. 7 163-186. · Zbl 0401.62031 · doi:10.1214/aos/1176344562
[9]	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. · Zbl 0718.60087 · doi:10.1214/aos/1176347865
[10]	Hjort, N. L. (1990). Nonparametric Bay es estimators based on beta processes in models for life history data. Ann. Statist. 18 1259-1294. · Zbl 0711.62033 · doi:10.1214/aos/1176347749
[11]	Lavine, M. (1992). Some aspects of Póly a tree distributions for statistical modelling. Ann. Statist. 20 1222-1235. · Zbl 0765.62005 · doi:10.1214/aos/1176348767
[12]	Lavine, M. (1994). More aspects of Póly a trees for statistical modelling. Ann. Statist. 22 1161- 1176. · Zbl 0820.62016 · doi:10.1214/aos/1176325623
[13]	Lévy, P. (1936). Théorie de l’Addition des Variables Aléatoire. Gauthiers-Villars, Paris.
[14]	Lo, A. Y. (1988). A Bayesian bootstrap for a finite population. Ann. Statist. 16 1684-1695. · Zbl 0691.62005 · doi:10.1214/aos/1176351061
[15]	Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Póly a trees and random distributions. Ann. Statist. 20 1203-1221. · Zbl 0765.62006 · doi:10.1214/aos/1176348766
[16]	Mihram, G. A. and Hultquist, R. A. (1967). A bivariate warning-time/failure-time distribution. J. Amer. Statist. Assoc. 62 589-599. JSTOR: · Zbl 0171.16802 · doi:10.2307/2283985
[17]	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. JSTOR: · Zbl 0907.62036 · doi:10.1111/1467-9868.00116
[18]	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. JSTOR: · Zbl 0779.62030
[19]	Susarla, V. and Van Ry zin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete data. J. Amer. Statist. Assoc. 71 897-902. JSTOR: · Zbl 0344.62036 · doi:10.2307/2286858