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

Abstract

beta-Stacy process is dened. 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 P olya-urn scheme is introduced which characterizes the discrete betaStacy process. 1. Introduction. Let F be the space of cumulative distribution functions (cdfs) oni0;1e. This paper considers placing a probability distribution on F by dening a stochastic process F onei0;1e; Ae, where A is the Borel eld of subsets, such that Fe0e D 0 a.s., F is a.s. nondecreasing, a.s. right continuous and lim t!1 Fete D 1 a.s. Thus, with probability 1, the sample paths of F are cdf’s. Previous work includes the Dirichlet process [Ferguson (1973, 1974)], neutral to the right processes [Doksum (1974)], the extended gamma process [Dykstra and Laud (1981)], the beta process [Hjort (1990)] and P olya trees [Lavine (1992, 1994), Mauldin, Sudderth and Williams (1992)]. The purpose of this paper is twofold: (1) to introduce a new stochastic process which generalizes the Dirichlet process, in that more exible 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 P olyaurn 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 present paper is restricted to considering the estimation of an unknown cdf oni0;1e, although it is trivially extended to includee1 ;1e. 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 dened and its connections with other processes given. We also provide an