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The sampling theory of selectively neutral alleles

Published online by Cambridge University Press:  01 July 2016

G. A. Watterson
G. A. Watterson*
Affiliation:
Monash University
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Abstract

In a paper of the same title, Ewens has considered the problem of inferring whether the genotypic frequencies observed in a small sample are consistent with a particular population model, in which all types are selectively neutral. The present paper gives the general theory of such sampling schemes, when the sampling is from either a deterministic, or a stochastically varying, population.

The theory is illustrated on a population model which was introduced by Karlin and McGregor. Various results are proved for both the population, and its sample. It is shown that Ewens' sampling theory applies to this population model, and conjectures as to why different population models may have the same sampling theory are made.

Keywords

SAMPLING OF TYPESOCCUPANCY PROBLEMGENETIC POPULATION MODELSESTIMATIONGOODNESS OF FITSELECTIVE NEUTRALITY HYPOTHESIS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 463 - 488
Copyright
Copyright © Applied Probability Trust 1974 

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References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
David, F. N. and Barton, D. E. (1962) Combinatorial Chance. Charles Griffin, London.Google Scholar
Ewens, W. J. (1964) The maintenance of alleles by mutation. Genetics 50, 891–898.Google Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theor. Pop. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Ewens, W. J. and Gillespie, J. H. (1974) Simulation results for the neutral allele model. To appear.Google Scholar
Guess, H. A. and Ewens, W. J. (1972) Theoretical and simulation results relating to the neutral allele theory. Theor. Pop. Biol. 3, 434–447.Google Scholar
Karlin, S. and McGregor, J. (1958) Linear growth, birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Karlin, S. and McGregor, J. (1967) The number of mutant forms maintained in a population. Proc. 5th Berkeley Symp. Math. Statist. Prob. IV, 415–438.Google Scholar
Karlin, S. and McGregor, J. (1972) Addendum to a paper of W. Ewens. Theor. Pop. Biol. 3, 113–116.Google Scholar
Kendall, D. G. (1948) On some modes of population growth leading to R. A. Fisher's logarithmic series distribution. Biometrika 35, 6–15.Google Scholar
Kimura, M. and Crow, J. (1964) The number of alleles that can be maintained in a finite population. Genetics 49, 725–738.Google Scholar
Riordan, J. (1958) An Introduction to Combinatorial Analysis. John Wiley, New York.Google Scholar
Singer, B. (1970) Some asymptotic results in a model of population growth. I. Ann. Math. Statist. 41, 115–132.Google Scholar
Singer, B. (1971) Some asymptotic results in a model of population growth. II. Ann. Math. Statist. 42, 1296–1315.CrossRefGoogle Scholar
Trajstman, A. C. (1974) On a conjecture of G. A. Wattersen. Adv. Appl. Prob. 6, 489–493.Google Scholar
Tukey, J. W. (1949) Moments of random group size distributions. Ann. Math. Statist. 20, 523–539.CrossRefGoogle Scholar
Watterson, G. A. (1974) Models for the logarithmic species abundance distributions. Theor. Pop. Biol. To appear.Google Scholar