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

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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. , 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