Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-01T11:15:01.966Z Has data issue: false hasContentIssue false
  • English
  • Français

Likelihood ratios for the infinite alleles model

Part of: Parametric inference Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Paul Joyce
Paul Joyce*
Affiliation:
University of Idaho
*
∗ Postal address: University of Idaho, Department of Mathematics and Statistics, Moscow, Idaho 83843, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The stationary distribution for the population frequencies under an infinite alleles model is described as a random sequence (x1, x2, · ··) such that Σxi = 1. Likelihood ratio theory is developed for random samples drawn from such populations. As a result of the theory, it is shown that any parameter distinguishing an infinite alleles model with selection from the neutral infinite alleles model cannot be consistently estimated based on gene frequencies at a single locus. Furthermore, the likelihood ratio (neutral versus selection) converges to a non-trivial random variable under both hypotheses. This shows that if one wishes to test a completely specified infinite alleles model with selection against neutrality, the test will not obtain power 1 in the limit.

Keywords

EXCHANGEABLE EQUIVALENCE RELATIONSMARTINGALES

MSC classification

Primary: 62F05: Asymptotic properties of tests
Secondary: 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 595 - 605
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Billingsley, P. (1985) Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1987) The infinitely-many-alleles model with selection as measure-valued diffusion. In Lecture Notes in Biomathematics 70, 72–86. Springer-Verlag, Berlin.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1991) Convergence to Fleming-Viot processes in the weak atomic topology. Preprint.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
Joyce, P. and Tavaré, S. (1994) The distribution of rare alleles. J. Math. Biol. To appear.Google Scholar
Kimura, M. (1983) Rare variant alleles in the light of the neutral theory. Molecular Biology and Evolution 1, 84–93.Google ScholarPubMed
Kingman, J. F. C. (1980) Mathematics of Genetic Diversity. SIAM, Philadelphia.CrossRefGoogle Scholar
Kingman, J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13, 235–248.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.CrossRefGoogle Scholar
Li, W.-H. (1978) Maintenance of genetic variability under the joint effect of mutation, selection and random drift. Genetics 90, 349–382.CrossRefGoogle ScholarPubMed
Li, W.-H. (1979) Maintenance of genetic variability under the pressure of neutral and deleterious mutations in a finite population. Genetics 92, 647–667.CrossRefGoogle Scholar
Watterson, G. A. (1978) The homozygosity test of neutrality. Genetics 88, 405–417.Google Scholar
Watterson, G. A. (1987) Estimating the proportion of neutral mutants. Genet. Res. Camb. 501, 155–163.CrossRefGoogle Scholar