MAXIMUM-LIKELIHOOD-ESTIMATION FOR GALTON-WATSON PROCESSES

A recursive formula is derived for the transition probabilities of a Galton-Watson branching process in which members of the population have at most K offspring. Expressions are found for the derivatives of these transition probabilities with respect to the parameters, p0, p1,..., p(K), that govern the probabilities of having 0,1,...,K offspring. The recursive formula and the expressions for the derivatives make it feasible to estimate the parameters of the offspring distribution by the method of maximum likelihood. For various processes with K-2 we compare the small-sample properties of maximum-likelihood estimators with those of "method-of-moments" estimators, which are derived from the usual consistent estimators of the mean and variance of numbers of offspring. The m.l.e.s are found to have smaller mean squared errors.