Isotonic Estimation of the Probability of Extinction of a Branching Process
Abstract Suppose X 1, X 2, ···, Xn 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.