Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process

Abstract

We consider a single-type supercritical or near-critical size-dependent branching process { N n } n such that the offspring mean converges to a limit m ⩾ 1 with a rate of convergence of order N n α as the population size N n grows to ∞ and the variance may change at the rate N n β , where α > 0 and − 1 ⩽ β < 1 . The offspring mean depends on an unknown parameter θ 0 that we estimate on the non-extinction set by using the conditional least squares method. We prove the strong consistency of the estimator of θ 0 as n → ∞ under some general conditions on the asymptotic behavior of the process. We also give its asymptotic distribution for a certain class of size-dependent branching processes. To cite this article: N. Lalam, C. Jacob, C. R. Acad. Sci. Paris, Ser. I 339 (2004).