With Peter Guttorp I once investigated problems of estimation in these models of generation sizes in family trees. The results we have here are mostly negative, showing the non-existence of consistent estimates of parameters of the offspring distribution other than the mean and variance. We have looked at the maximum likelihood estimate of the variance of the offspring distribution. Though this offspring distribution is not consistently estimable its mean and variance are. It is known that mle of the mean is consistent and we think we can prove that the mle of the variance is also consistent. Our proof relies on a uniform approximation of the likelihood based on a uniform version of the local central limit theorem.

Lockhart, R. A. (1982). On the non-existence of consistent estimates in Galton-Watson processes. J. Appl. Prob., 19 842--846.

Guttorp, P. and Lockhart, R.A. (1989). Estimation in sparsely sampled random walks. Stoch. Proc. Their Appl., 31 315--320.
Draft paper

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