Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T18:33:33.951Z Has data issue: false hasContentIssue false
  • English
  • Français

Partition structures and sufficient statistics

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, ID 83843, USA. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Is the Ewens distribution the only one-parameter family of partition structures where the total number of types sampled is a sufficient statistic? In general, the answer is no. It is shown that all counterexamples can be generated via an urn scheme. The urn scheme need only satisfy two general conditions. In fact, the conditions are both necessary and sufficient. However, in particular, for a large class of partition structures that naturally arise in the infinite alleles theory of population genetics, the Ewens distribution is the only one in this class where the total number of types is sufficient for estimating the mutation rate. Finally, asymptotic sufficiency for parametric families of partition structures is discussed.

Keywords

Infinite alleles modelexchangeable equivalence relationssufficient statisticspartition structures

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 35 , Issue 3 , September 1998 , pp. 622 - 632
Copyright
Copyright © Applied Probability Trust 1998 

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 96–26764.

References

Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Limit theorems for combinatorial structures via discrete process approximations. Rand. Struct. Alg. 3, 321–345.Google Scholar
Donnelly, P. (1986). Partition structures, Poĺya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Popn Biol. 30, 271–288.Google Scholar
Donnelly, P., and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. Appl. Prob. 23, 229–258.Google Scholar
Donnelly, P., and Joyce, P. (1992). Weak convergence of population genealogical processes to a coalescent with ages. Ann. Prob. 20, 322–341.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1987). The infinitely-many-alleles model with selection as a measure-valued diffusion. Lecture Notes in Biomathematics 70. Springer, Berlin, pp. 72–86.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1994). Convergence to Fleming–Viot processes in the weak atomic topology. Stoch. Proc. Appl. 54, 1–27.Google Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.Google Scholar
Ewens, W. J. (1996). Remarks on the Law of Succession. In Athens Conference on Applied Probability and Time Series, Vol. I. Ed. Heyde, C. C., Prohorov, Y. V., Rachev, S. T. and Pyke, R. Lecture Notes in Statistics 114, Springer, Berlin, p. 229–244.Google Scholar
Griffiths, R. C. (1983). Allele frequencies with genic selection. J. Math. Biol. 17, 1–10.Google Scholar
Grote, M. (1996). Sampling and inference at a locus under symmetric overdominance selection. , Dept. of Integrative Biology, University of California, Berkeley.Google Scholar
Hoppe, F. M. (1984). Poĺya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–99.Google Scholar
Joyce, P. (1994). Likelihood ratios for the infinite alleles model. J. Appl. Prob. 31, 595–605.Google Scholar
Joyce, P. (1995). Robustness of the Ewens sampling formula. J. Appl. Prob. 32, 609–622.Google Scholar
Joyce, P. and Tavaré, S. (1995). The distribution of rare alleles. J. Math. Biol. 33, 602–618.Google Scholar
Kingman, J. F. C. (1978a). Random partitions in population genetics. Proc. R. Soc. London A 361, 1–20.Google Scholar
Kingman, J. F. C. (1978b). The representation of partition structures. J. London Math. Soc. 18, 374–380.Google Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. J. Appl. Prob. 19A, 27–43.Google Scholar
Kimura, M. (1983). Rare variant alleles in the light of the neutral theory. Mol. Biol. Evol. 1, 84–93.Google Scholar
Pitman, J. (1992). The two-parameter generization of Ewens' random partition structure. Technical Report 345, Department of Statistics, University of California at Berkeley.Google 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.Google Scholar
Watterson, G. A. (1993). Estimating the proportion of neutral mutations. New Zealand J. Botany 31, 297–306.Google Scholar
Zabell, S. L. (1996). The continuum of inductive methods revisited. In The Cosmos of Science. Ed. Earman, J. and Norton, J. University of Pittsburgh Series in the History and Philosophy of Science.Google Scholar