Optimal estimation of the criticality parameter of a supercritical branching process having random environments

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Abstract
Let {Zn} be a Galton-Watson process with i.i.d. random environments. This paper is concerned with estimation of π= E log Mn, where Mn is the. conditional expected number of offspring per nth generation individual, given the environments, when this quantity is positive. We show that, given non-extinction, {n-1 log Zn} is asymptotically the most efficient estimator of πamongst a broad class of those that are linear combinations of {log Zi: 1 ≦ j ≦ n}·

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