Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T10:07:28.703Z Has data issue: false hasContentIssue false
  • English
  • Français

Random discrete distributions invariant under size-biased permutation

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jim Pitman
Jim Pitman*
Affiliation:
University of California at Berkeley
*
∗ Postal address: Department of Statistics, U.C. Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.

Keywords

RESIDUAL ALLOCATION MODELEXCHANGEABLE RANDOM PARTITIONGENERALIZED DIRICHLET DISTRIBUTION

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60G09: Exchangeability

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 525 - 539
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Article purchase

Temporarily unavailable

Footnotes

Research supported by N.S.F. Grants MCS91–07531 and DMS-9404345.

References

[1] Aldous, D. J. (1985) Exchangeability and related topics. In Ecole d'Été de Probabilités de Saint-Flour XII (Springer Lecture Notes in Mathematics 1117) ed. Hennequin, P. L. Springer, Berlin.Google Scholar
[2] Connor, R. J. and Mosimann, J. E. (1969) Concepts of independence for proportions with a generalization of the Dirichlet distribution. J. Amer. Statist. Assoc. 64, 194–206.Google Scholar
[3] Donnelly, P. (1991) The heaps process, libraries and size biased permutations. J. Appl. Prob. 28, 322–335.Google Scholar
[4] Donnelly, P. and Joyce, P. (1989) Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stoch. Proc. Appl. 31, 89–103.Google Scholar
[5] Engen, S. (1978) Stochastic Abundance Models with Emphasis on Biological Communities and Species Diversity. Chapman and Hall, London.CrossRefGoogle Scholar
[6] Ewens, W. J. (1988) Population genetics theory–the past and the future. In Mathematical and Statistical Problems in Evolution. ed. Lessard, S. University of Montreal Press, Montreal.Google Scholar
[7] Fabius, J. (1964) Asymptotic behavior of Bayes estimates. Math. Statist. 35, 846–856.CrossRefGoogle Scholar
[8] Freedman, D. (1963) On the asymptotic behavior of Bayes estimates in the discrete case. Ann. Math. Statist. 34, 1386–1403.Google Scholar
[9] Hoppe, F. M. (1986) Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics. J. Appl. Prob. 23, 1008–1012.CrossRefGoogle Scholar
[10] Ignatov, T. (1982) On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures. Theory Prob. Appl. 27, 136–147.Google Scholar
[11] Kingman, J. F. (1975) Random discrete distributions. J. R. Statist. Soc. B 37, 1–22.Google Scholar
[12] Kingman, J. F. (1978) The representation of partition structures. J. London Math. Soc. 18, 374–380.CrossRefGoogle Scholar
[13] Kingman, J. F. (1982) The coalescent. Stoch. Proc. Appl. 13, 235–248.Google Scholar
[14] Mccloskey, J. W. (1965) A model for the distribution of individuals by species in an environment. Ph.D. thesis., Michigan State University.Google Scholar
[15] Patil, G. P. and Taillie, C. (1977) Diversity as a concept and its implications for random communities. Bull. Int. Statist. Inst. XLVII, 497–515.Google Scholar
[16] Perman, M., Pitman, J. and Yor, M. (1992) Size-biased sampling of Poisson point processes and excursions. Prob. Theory Rel. Fields 92, 21–39.Google Scholar
[17] Pitman, J. (1995) Exchangeable and partially exchangeable random partitions. Prob. Theory Rel. Fields 102, 145–158.CrossRefGoogle Scholar
[18] Pitman, J. (1992) The two-parameter generalization of Ewens' random partition structure. Technical Report 345. Dept. Statistics, U.C. Berkeley.Google Scholar
[19] Pitman, J. and Yor, M. (1996) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Prob. (to appear).Google Scholar
[20] Pitman, J. and Yor, M. (1996) Random discrete distributions derived from self-similar random sets. Electronic J. Prob. 1, Paper 4. (http://math.washington.edu/∽ejpecp/EjpVol1/paper4.abs.html)Google Scholar
[21] Watterson, G. A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.Google Scholar
[22] Zabell, S. L. (1992) Predicting the unpredictable. Synthese 90, 205–232.CrossRefGoogle Scholar
[23] Zabell, S. L. (1996) The continuum of inductive methods revisited. In The Cosmos of Science. (Pittsburgh-Konstanz Series in the Philosophy and History of Science). ed. Earman, J. and Norton, J. University of Pittsburgh Press/Universitats verlag Konstanz.Google Scholar