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

Abstract

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β < 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.