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Approximate Ewens formulae for symmetric overdominance selection. (English) Zbl 1010.62100
From the paper: Ewens distributions [see W.J. Ewens, Mathematical population genetics. (1979; Zbl 0422.92011)] arise naturally in a number of sampling problems in the biological and physical sciences. In population genetics, Ewens distributions have been used to describe samples at genetic loci which follow the “infinitely-many neutral alleles” model. The basic features of this model are: (i) alleles (distinct versions of a gene) are equivalent with respect to natural selection, and (ii) mutation, which occurs in any gene copy with probability $$u$$ each generation, transforms the copy into a completely new allele.
We derive a family of approximate sampling distributions for the symmetric overdominance model of population genetics. The distributions are selective versions of the Ewens Sampling Formula, which gives sample likelihoods under a model of neutral evolution. We draw on basic results for the general selection model of S.N. Ethier and T.G. Kurtz [see Lect. Notes Biomath. 70, 72-86 (1987; Zbl 0644.92011)] and use mathematical tools well-suited for calculating expectations of symmetric functions of Poisson-Dirichlet atoms. We conclude by briefly examining a Human Leukocyte Antigen data set, in light of a distribution conditional on the number of sample atoms.

##### MSC:
 62P10 Applications of statistics to biology and medical sciences; meta analysis 92D15 Problems related to evolution 92D10 Genetics and epigenetics
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