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In this paper, we construct an approximation to the Pitman–Yor process by truncating its two‐parameter Poisson–Dirichlet representation. The truncation is based on a decreasing sequence of random weights, thus having a lower approximation error compared to the popular truncated stick‐breaking process. We develop an exact simulation algorithm to sample from the approximation process and provide an alternative MCMC algorithm for the parameter regime where the exact simulation algorithm becomes slow. The effectiveness of the simulation algorithms is demonstrated by the estimation of the functionals of a Pitman–Yor process. Then we adapt the approximation process into a Pitman–Yor process mixture model and devise a blocked Gibbs sampler for posterior inference. [ABSTRACT FROM AUTHOR]

In forensic science, the rare type match problem arises when the matching characteristic from the suspect and the crime scene is not in the reference database; hence, it is difficult to evaluate the likelihood ratio that compares the defense and prosecution hypotheses. A recent solution consists of modeling the ordered population probabilities according to the two‐parameter Poisson–Dirichlet distribution, which is a well‐known Bayesian nonparametric prior, and plugging the maximum likelihood estimates of the parameters into the likelihood ratio. We demonstrate that this approximation produces a systematic bias that fully Bayesian inference avoids. Motivated by this forensic application, we consider the need to learn the posterior distribution of the parameters that governs the two‐parameter Poisson–Dirichlet using two sampling methods: Markov Chain Monte Carlo and approximate Bayesian computation. These methods are evaluated in terms of accuracy and efficiency. Finally, we compare the likelihood ratio that is obtained by our proposal with the existing solution using a database of Y‐chromosome haplotypes. [ABSTRACT FROM AUTHOR]

In this paper three models are considered. They are the infinitely-many-neutral-alleles model of Ethier and Kurtz (1981), the two-parameter infinitely-many-alleles diffusion model of Petrov (2009), and the infinitely-many-alleles model with symmetric dominance Ethier and Kurtz (1998). New representations of the transition densities are obtained for the first two models and the ergodic inequalities are provided for all three models. [ABSTRACT FROM AUTHOR]

In this paper we investigate the relationship between the sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution. We conclude that they are equivalent to determining the corresponding infinite-dimensional distribution. With these tools, a central limit theorem is established associated with the infinitely-many-neutral-alleles model at any fixed time. We also obtain the probability generating function of random sampling from a generalized two-parameter diffusion process. At the end of the paper a selection case is considered. [ABSTRACT FROM AUTHOR]

Abstract: By explicitly identifying the transition density function, we derived the super-Poincaré and super-log-Sobolev inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion, which in particular implies the Gross log-Sobolev inequality. [ABSTRACT FROM AUTHOR]

Abstract: The two-parameter Poisson–Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and gamma subordinators with the two parameters, and , corresponding to the stable component and the gamma component respectively. The moderate deviation principle is established for the distribution when approaches infinity, and the large deviation principle is established when both and approach zero. [Copyright &y& Elsevier]

In this paper we introduce the transformed two-parameter Poisson-Dirichiet distribution ⨅σθ,α on the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rate θ and the selection rate σ become large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength of σ and θ. [ABSTRACT FROM AUTHOR]