Isotonic Estimation of the Probability of Extinction of a Branching Process

Abstract

Suppose X ₁, X ₂, ···, X_n is a random sample from a distribution which assigns all of its mass to the nonnegative integers. Let p _j = P[X ₁ = j]; j = 0, 1, ··· and let f(·) be the generating function of the sequence {p _j }^∞ _j=0. An estimate of the smallest nonnegative root of f(s) = s which is based on isotonized estimates of p ₀, p ₁, ··· is considered. Results of a simulation study, almost sure convergence rates and the asymptotic distribution theory are discussed.