Asymptotic Conditional Inference for the Offspring Mean of a Supercritical Galton-Watson Process

Abstract

Consider a supercritical Galton-Watson process $(Z_n)$ with offspring distribution a member of the power series family, and having unknown mean $\theta$. The conditional asymptotic normality of the suitably normalized maximum likelihood estimator of $\theta$ given the conditional information is established. The conditional information here is proportional to the total number of ancestors $V_n$, and it is also seen that this statistic is asymptotically ancillary for $\theta$ in a local sense. The proofs are via a detailed analysis of the joint characteristic function of $(Z_n, V_n)$, and the derivation serves to highlight the difficulties involved in establishing such conditional results generally.