Quasi-equilibrium theory for the distribution of rare alleles in a subdivided population: Justification and implications.

*(English)*Zbl 0974.92021Summary: This paper examines a quasi-equilibrium theory of rare alleles for subdivided populations that follow an island-model version of the Wright-Fisher model of evolution. All mutations are assumed to create new alleles. We present four results:

(1) conditions for the theory to apply are formally established using properties of the moments of the binomial distribution; (2) approximations currently in the literature can be replaced with exact results that are in better agreement with our simulations; (3) a modified maximum likelihood estimator of migration rate exhibits the same good performance on island-model data or on data simulated from the multinomial mixed with the Dirichlet distribution, and (4) a connection between the rare-allele method and the Ewens Sampling Formula [W.J. Ewens, Theor. Popul. Biol. 3, 87-112 (1972; Zbl 0245.92009)]; Mathematical population genetics. (1979; Zbl 0422.92011)] for the infinite-allele mutation model is made. This introduces a new and simpler proof for the expected number of alleles implied by the Ewens Sampling Formula.

(1) conditions for the theory to apply are formally established using properties of the moments of the binomial distribution; (2) approximations currently in the literature can be replaced with exact results that are in better agreement with our simulations; (3) a modified maximum likelihood estimator of migration rate exhibits the same good performance on island-model data or on data simulated from the multinomial mixed with the Dirichlet distribution, and (4) a connection between the rare-allele method and the Ewens Sampling Formula [W.J. Ewens, Theor. Popul. Biol. 3, 87-112 (1972; Zbl 0245.92009)]; Mathematical population genetics. (1979; Zbl 0422.92011)] for the infinite-allele mutation model is made. This introduces a new and simpler proof for the expected number of alleles implied by the Ewens Sampling Formula.

##### MSC:

92D15 | Problems related to evolution |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

##### Keywords:

Ewens sampling formula; Wright-Fisher model; binomial distribution; maximum likelihood estimator
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\textit{T. L. Burr}, Theor. Popul. Biol. 57, No. 3, 297--306 (2000; Zbl 0974.92021)

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##### References:

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