Recurrence formula and the maximum likelihood estimation of the age in a simple branching process

Abstract

In this paper, we consider a Bienaymé– Galton–Watson process {Xn; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.