Estimation of the support of a discrete distribution

Abstract

Let Y be a positive integer-valued random variable with the probability mass function P θ(Y=y)=f(y;r)/a(θ), y=r,r+1,…,θ , where r is a known positive integer, and θ∈Θ={r,r+1,…} is an unknown parameter. We show that, for estimating θ, cY is inadmissible under both 0–1 and a general loss whenever 0<c<1. Under some mild conditions on f(y;r), we prove that Y is admissible and minimax under both 0–1 and squared error loss. As an application, we consider the problem of estimating the size θ of a finite population whose elements are labeled from 1 to θ, based on a simple random sample of size n under both with and without replacement. Admissibility and minimaxity of Y, the largest number observed in the sample, under 0–1 and squared error loss hold under both sampling situations. We propose two integer-valued estimators of θ of the form [cY] for c>1 in the case of sampling with replacement and discuss their bias and mean-squared error ( [cY] denotes the integer nearest to cY).