Minimum Hellinger Distance Estimation for Supercritical Galton-Watson Processes

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.