Sampling Formulae for Symmetric Selection

Kenji Handa (Saga University)

Abstract


We study partition distributions in a population genetics model incorporating symmetric selection and mutation. They generalize Ewens distributions in the infinitely-many-neutral-alleles model, an explicit expression of which is known as the Ewens sampling formula. A sampling formula for the generalized model is obtained by means of calculus for Poisson and gamma processes.

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Pages: 223-234

Publication Date: November 14, 2005

DOI: 10.1214/ECP.v10-1159

References

  1. R. Arratia, A. D. Barbour, S. Tavaré. Logarithmic Combinatorial Structures: a Probabilistic Approach. European Mathematical Society, Zurich, 2003. MR2032426 (2004m:60004)
  2. P. Billingsley. Convergence of Probability Measures. 2nd edition. John Wiley & Sons, Inc., New York, 1999. MR1700749 (2000e:60008)
  3. W. J. Ewens. The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 (1972), 87-112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376. MR0325177 (48 #3526)
  4. W. J. Ewens, S. Tavaré. Multivariate Ewens distribution. in: N. Johnson, S. Kotz, N. Balakrishnan (Eds.). Discrete Multivariate Distributions. John Wiley & Sons, Inc., New York, 1997, pp. 232-246. Math. Review number not available.
  5. M. N. Grote, T. P. Speed. Approximate Ewens formulae for symmetric overdominance selection. Ann. Appl. Probab. 12 (2002), 637-663. MR1910643 (2003c:62020)
  6. L. F. James. Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. preprint, 2002. available at http://front.math.ucdavis.edu/math.PR/0205093
  7. L. F. James. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statistics 33 (2005), 1771-1799.
  8. J. F. C. Kingman. Random discrete distribution. J. Roy. Statist. Soc. Ser. B 37 (1975), 1-22. MR0368264 (51 #4505)
  9. J. F. C. Kingman. Random partitions in population genetics. Proc. Roy. Soc. London Ser. A 361 (1978), 1-20. MR0526801 (58 #26167)
  10. J. F. C. Kingman. The representation of partition structures. J. London Math. Soc. (2) 18 (1978), 374-380. MR0509954 (80a:05018)
  11. J. F. C. Kingman. Poisson Processes. Oxford University Press, New York, 1993. MR1207584 (94a:60052)
  12. A. Y. Lo, C.-S. Weng. On a class of Bayesian nonparametric estimates, II, Hazard rate estimates. Ann. Inst. Statist. Math. 41 (1989), 227-245. MR1006487 (90j:62149)
  13. J. Pitman. Poisson-Kingman partitions. preprint, 1995.
  14. J. Pitman. Combinatorial stochastic processes. Technical Report No. 621, Dept. Statistics., U. C. Berkeley, 2002; Lecture notes for St. Flour course, July 2002. available at http://www.stat.berkeley.edu/users/pitman
  15. J. Pitman. Poisson-Kingman partitions. in: D. R. Goldstein (Ed.). Science and Statistics: A Festschrift for Terry Speed. Institute of Mathematical Statistics Hayward, California, 2003, pp. 1-34. MR2004330 (2004j:60019)
  16. J. Pitman, M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), 855-900. MR1434129 (98f:60147)
  17. N. Tsilevich, A. Vershik, M. Yor. Distinguished properties of the gamma process, and related topics. Prépublication du Laboratoire de Probabilités et Modèles Aléatoires. No. 575, 2000. available at http://xxx.lanl.gov/ps/math.PR/0005287
  18. N. Tsilevich, A. Vershik, M. Yor. An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. Journ. Funct. Anal. 185 (2001), 274-296. MR1853759 (2002g:46071)
  19. G. A. Watterson. Heterosis or neutrality ? Genetics 85 (1977), 789-814. MR0504021 (58 #20595)
  20. G. A. Watterson. The homozygosity test of neutrality. Genetics 88 (1978), 405-417. Math. Review number not available.


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