Application of the Galton-Watson process to the kin number problem

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Abstract
The kin number problem concerns the relationship between the distribution of the number of offspring of a randomly chosen individual in a population, and that of the number of relatives of various degrees of affinity of a randomly chosen individual (referred to as ‘Ego’). This problem is considered in terms of a population consisting of a large number of simultaneously developing Galton-Watson family trees. By time-reversal starting with the epoch at which Ego is sampled, it is shown that the number of offspring of Ego's parent (Ego plus siblings) and the number of offspring of Ego's grandparent (Ego's parent plus Ego's parent's siblings) have the same distribution, and the probability generating function (p.g.f.) is obtained in terms of the reproduction p.g.f. Further development of the method yields the joint p.g.f. of any number of generations prior, and subsequent, to that of Ego. Means and variances are obtained, and numerical examples are given based on data for the reproduction p.g.f. in two contrasting human populations. Various applications including demography and cancer research are discussed.