The kin number problem in a multitype Galton–Watson population

Abstract

The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), Joffe and Waugh (1982)).In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken. The existence of more than one type complicates the time-reversal used in the previous studies, and raises questions about the way in which the focal individual (called ‘Ego') is sampled. These are dealt with, and joint distributions obtained, under a number of sampling schemes which might arise naturally.