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Exact distributions of kin numbers in a Galton-Watson process

Published online by Cambridge University Press:  14 July 2016

A. Joffe  and
W. A. O'n. Waugh
A. Joffe*
Affiliation:
Université de Montréal
W. A. O'n. Waugh*
Affiliation:
University of Toronto
*
∗ Postal address: Centre de Recherche de Mathématiques Appliquées, Université de Montréal, P.O. Box 6128, Succ. A, Montréal. Québec H3C 3J7, Canada.
∗∗ Postal address: Department of Statistics, The University of Toronto, Sidney Smith Hall, Toronto, Ontario M5S 1A1, Canada.
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Abstract

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations.

In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.

Keywords

KIN NUMBER PROBLEMBRANCHING POPULATION MODELSMINI-DEMOGRAPHYCONDITIONED GALTON-WATSON PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 4 , December 1982 , pp. 767 - 775
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Research supported in part by operating grants from the Natural Sciences and Engineering Research Council, Canada and F.C.A.C. programme of the Ministère de l'Education du Québec.

References

Burks, B. S. (1933) A statistical method for estimating the distribution of sizes of completed fraternities in a population represented by a random sampling of individuals. J. Amer. Statist. Assoc. 28, 388–394.Google Scholar
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Jagers, P. (1982) How probable is it to be firstborn? and other branching process applications to kinship problems. Math. Biosci. 59, 1–16.Google Scholar
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Waugh, W.A.O'N. (1981) Application of the Galton-Watson process to the kin number problem. Adv. Appl. Prob. 13, 631–649.Google Scholar