Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula

Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $\theta$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $\theta$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $\theta$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson--Dirichlet distribution in terms of large deviations. Large $\theta$ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.Comment: Published in at http://dx.doi.org/10.1214/105051607000000230 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic behavior of the Poisson--Dirichlet distribution for large mutation rate

The large deviation principle is established for the Poisson--Dirichlet distribution when the parameter $\theta$ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson--Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in $\theta$, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in $\theta$, then the large deviation rate function includes an additional term coming from selection. The application of these results to the heterozygote advantage model provides an alternate proof of one of Gillespie's conjectures in [Theoret. Popul. Biol. 55 145--156].Comment: Published at http://dx.doi.org/10.1214/105051605000000818 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large deviation principles for the Ewens-Pitman sampling model

Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional sample of size $m$. For any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly, the conditional and unconditional large deviation principles coincide, namely there is no long lasting impact of the given initial sample. Potential applications of our results are discussed in the context of Bayesian nonparametric inference for discovery probabilities.Comment: 30 pages, 2 figure