The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...

Le"vy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 1.

Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, two-parameter Poisson-Dirichlet process, prediction rule, random probability measure, species sampling sequence.

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].

We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures.

Summary. In this paper we consider the component structure of decomposable combi-natorial objects, both labeled and unlabeled, from a probabilistic point of view. In both cases we show that when the generating function for the components of a structure is a logarithmic function, then the joint distribution of the normalized order statistics of the component sizes of a random object of size n converges to the Poisson-Dirichlet distribu-tion on the simplex ∇ = {{xi} : � xi =1,x1 ≥ x2 ≥... ≥ 0}. This result complements recent results obtained by Flajolet and Soria [9] on the total number of components in a random combinatorial structure.

this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stick-breaking measures, which includes establishing a conjugacy property of their random weights to i.i.d sampling, and a characterization of the posterior for the extended stick-breaking prior under i.i.d sampling. See Section 3. These properties then lead us in Section 4 to a general characterization for the posterior of (7). In Section 5 we outline a collapsed Gibbs sampling algorithm and an i.i.d SIS (sequential importance sampling) algorithm that can be used for inference in (7). One important implication is our ability to t the posterior of (6) subject to in nite dimensional stick-breaking measures. The paper begins with a brief discussion of stick-breaking priors in Section 2

This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the two-parameter family of Poisson-Dirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.

Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavy-tailed distributions in the study of limiting processes, while in Bayesian nonparametric statistics they are used as constructive representations for nonparametric priors. We describe a simple recursive method that is useful for characterizing Poisson process functionals that requires only the use of conditional probability. Applications of this technique to convex hulls, extremes, stable measures, infinitely divisible random variables and Bayesian nonparametric priors are discussed.

The prevalence of categorical observations that are unique in a sample and also still unique in the population are usually taken as the measure of the overall risk of disclosure in the sample data. Samuels (1998) suggested adopting evolutionary processes and their associated urn models as a framework for estimating this prevalence. We re-examine his proposal and suggest several extensions that arise naturally in the Bayesian statistical framework. We provide a brief report on some empirical studies using data provided by the Israel Central Bureau of Statistics. We also link this approach to ones based on the structure of cross-classifications allowing for differential, per-unit forms of risk assessment. 1