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Robustness of the Ewens sampling formula

Part of: Parametric inference Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Paul Joyce
Paul Joyce*
Affiliation:
University of Idaho
*
∗Postal address: Department of Mathematics and Statistics, University of Idaho, Moscow, Idaho 83844, USA.
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Abstract

Under the assumptions of the neutral infinite alleles model, K (the total number of alleles present in a sample) is sufficient for estimating θ (the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator for θ based on K. It is shown that the same estimator used to estimate θ under neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.

Keywords

INFINITE ALLELES MODELEXCHANGEABILITYABSOLUTE CONTINUITYTAIL SIGMA FIELDSRESIDUAL ALLOCATION MODELS

MSC classification

Primary: 62F05: Asymptotic properties of tests
Secondary: 60G42: Martingales with discrete parameter 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 609 - 622
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

This research is supported by the National Science Foundation, DMS 92-07410.

References

Aldous, D. J. (1985) Exchangeability and related topics, In école d'été de Probabilités de Saint-Flour XIII-1983, pp. 2–198. Lecture Notes in Mathematics 1117, Springer-Verlag, Berlin.Google Scholar
Arratia, R., Barbour, A. D. and Tavare, S. (1992) Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519–535.Google Scholar
Billingsley, P. (1985) Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
Donnelly, P, and Joyce, P. (1991) Consistent ordered sampling distribution characterization and convergence. Adv. Appl. Prob. 23, 229–258.Google Scholar
Donnelly, P., Kurtz, T. G. and Tavaré, S. (1991) On the functional central limit theorem for the Ewens sampling formula. Ann. Appl. Prob. 1, 539–545.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1987) The infinitely-many-alleles model with selection as a measurevalued diffusion. Lecture Notes in Biomathematics 70, pp. 72–86. Springer-Verlag, Berlin.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1993) Convergence to Fleming-Viot processes in the weak atomic topology. Preprint.CrossRefGoogle Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn. Biol. 3, 87–112.Google Scholar
Ewens, W. J. (1979) Mathematical Population Genetics. Springer-Verlag, Berlin.Google Scholar
Ewens, W. J. and Li, W.-H. (1980) Frequency spectra of neutral and deleterious alleles in a finite population. J. Math. Biol. 10, 155–166.Google Scholar
Griffiths, R. C. (1983) Allele frequencies with genic selection. J. Math. Biol. 17, 1–10.Google Scholar
Halmos, P. R. (1944) Random alms. Ann. Math. Statist. 15, 182–189.Google Scholar
Hansen, J. C. (1990) A functional central limit theoerem for the Ewens sampling formula. J. Appl. Prob. 27, 28–43.Google Scholar
Joyce, P. (1994) Likelihood ratios and the infinite alleles model. J. Appl. Prob. 31, 595–605.Google Scholar
Joyce, P. and Tavaré, S. (1994) The distribution of rare alleles. J. Math. Biol. Submitted.Google Scholar
Kingman, J. F. C. (1982a) On the genealogy of large populations. J. Appl. Prob. 19A, 27–413.Google Scholar
Kingman, J. F. C. (1982b) The coalescent. Stoch. Proc. Appl. 13, 235–248.CrossRefGoogle Scholar
Kingman, J. F. C. (1982c) Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, ed. Koch, G. and Spizzichino, F., pp. 97–112. North-Holland, Amsterdam.Google Scholar
Li, W.-H. (1977) Maintenance of genetic variability under mutation and selection pressures in a finite population. Proc. Natl. Acad. Sci. USA 74, 2509–2513.Google Scholar
Li, W.-H. (1978) Maintenance of genetic variability under the joint effect of mutation, selection and random drift. Genetics 90, 349–382.Google Scholar
Li, W.-H. (1979) Maintenance of genetic variability under the pressure of neutral and deleterious mutations in a finite population. Genetics 92, 647–667.Google Scholar
Watterson, G. A. (1974) The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.CrossRefGoogle Scholar
Watterson, G. A. (1978) The homozygosity test of neutrality. Genetics 88, 405–417.Google Scholar
Wright, S. (1949) In Adaptation and Selection in Genetics, ed. Jepson, C. L., Simpson, C. G. and Mayr, E. Princeton Univeristy Press.Google Scholar