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 nonexistence 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 nonexistence of consistent estimates in
GaltonWatson processes. J. Appl. Prob., 19 842846.
Guttorp, P. and Lockhart, R.A. (1989). Estimation in sparsely
sampled random walks. Stoch. Proc. Their Appl., 31 315320.
Draft paper
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