On Estimating the Variance of the Offspring Distribution in a Simple Branching Process

Abstract

A single realization {Z n , 0 ≦n≦ N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator \(N^{ - 1} \sum {_{j = 1}^N } \left( {Z_{j + 1} - \hat mZ_j } \right)^2 Z_j^{ - 1} \) is proposed, where \(\hat m = Z_{N + 1} /Z_N \), and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.