Beta-stacy processes and a generalization of the Pólya-Urn scheme
type: Item
instance of: scholarly article
title: Beta-stacy processes and a generalization of the Pólya-Urn scheme
zbMATH Open document ID: 0928.62067
author: Pietro Muliere
author: Stephen G. Walker
published in: The Annals of Statistics
publication date: 2000-01-11T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 62C10
Mathematics Subject Classification ID: 60G09
zbMATH DE Number: 1082930
zbMATH Keywords: Bayesian nonparametrics
zbMATH Keywords: generalized Dirichlet distribution
zbMATH Keywords: generalized Polya-urn scheme
zbMATH Keywords: Levy process
zbMATH Keywords: neutral to the right process
zbMATH Keywords: Dirichlet process
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1031594741
OpenAlex ID: W1984455046
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DOI: 10.1214/AOS/1031594741
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description: scientific article; zbMATH DE number 1082930