Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.
%0 Journal Article
%1 lessard2010recurrence
%A Lessard, Sabin
%D 2010
%J J. Appl. Probab.
%K Moran_model ewens_sampling_formula recurrence_relations sampling_formula wright_fisher_model
%N 3
%P 732--751
%R 10.1239/jap/1285335406
%T Recurrence equations for the probability distribution of
sample configurations in exact population genetics models
%U http://dx.doi.org/10.1239/jap/1285335406
%V 47
%X Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.
@article{lessard2010recurrence,
abstract = {Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough. },
added-at = {2011-03-10T20:30:00.000+0100},
author = {Lessard, Sabin},
biburl = {https://www.bibsonomy.org/bibtex/239d6302e1334c5a5762603075c8ef8b6/peter.ralph},
coden = {JPRBAM},
doi = {10.1239/jap/1285335406},
fjournal = {Journal of Applied Probability},
interhash = {c175e85d86f6b95ac5d657a05b4a8b21},
intrahash = {39d6302e1334c5a5762603075c8ef8b6},
issn = {0021-9002},
journal = {J. Appl. Probab.},
keywords = {Moran_model ewens_sampling_formula recurrence_relations sampling_formula wright_fisher_model},
mrclass = {60C05 (92Dxx)},
mrnumber = {2731345},
number = 3,
pages = {732--751},
timestamp = {2011-03-31T06:56:35.000+0200},
title = {Recurrence equations for the probability distribution of
sample configurations in exact population genetics models},
url = {http://dx.doi.org/10.1239/jap/1285335406},
volume = 47,
year = 2010
}