973 research outputs found
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 approaches infinity. The motivation for
these results comes from a desire of understanding the exact meaning of
going to infinity. In terms of the law of large numbers and the
central limit theorem, the limiting procedure of 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 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 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 , 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 , 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 be the number of blocks with frequency in the exchangeable
random partition induced by a sample of size from the Ewens-Pitman sampling
model. We show that, as tends to infinity, 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 from the Ewens-Pitman
sampling model, we consider an additional sample of size . For any fixed
and as tends to infinity, we establish a large deviation principle for the
conditional number of blocks with frequency 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
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