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Consistent ordered sampling distributions: characterization and convergence

Published online by Cambridge University Press:  01 July 2016

Peter Donnelly  and
Paul Joyce
Peter Donnelly*
Affiliation:
Queen Mary and Westfield College, London
Paul Joyce*
Affiliation:
University of Southern California
*
∗Postal address: School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, UK.
∗∗Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA.
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Abstract

This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.

Keywords

EXCHANGEABILITYWEAK CONVERGENCERANDOM RANKINGSPARTITION STRUCTURESGEM DISTRIBUTIONAGE-ORDERED ALLELE FREQUENCIESPOISSON–DIRICHLET DISTRIBUTIONGENEALOGY
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 229 - 258
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Research partially supported by NSF grant DMS 86-08857.

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