Minimum Hellinger Distance Estimation for Supercritical Galton-Watson Processes

Abstract

For supercritical branching processes, suppose we postulate that the probability mass function of the offspring variable belongs to a specified parametric family {f_q; q Î Q}. If the postulated model is correct, then one can use the MLE, q_MLE, to estimate theta it is known that hat q_MLE is asymptotically efficient. However, it is also well known that MLEs do not, in general, possess the property of stability under small perturbations in the underlying model. To overcome this deficiency, we propose a minimum Hellinger distance estimator hat q_HD. We establish its existence, uniqueness, consistency and derive the limit distribution. It is shown that (a) hat q_HD is asymptotically efficient (just as the MLE is) if the postulated model is in fact true and (b) the limit distribution of hat q_HD is not greatly perturbed if the assumed model is only approximately true. Robustness aspects such as the influence curve and the asymptotic breakdown point of hat q_HD are also derived.

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