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

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.

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
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\textit{S. Walker} and \textit{P. Muliere}, Ann. Stat. 25, No. 4, 1762--1780 (1997; Zbl 0928.62067)

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