Estimation of the support of a discrete distribution

Let Y be a positive integer-valued random variable with the probability mass function P[theta](Y=y)=f(y;r)/a([theta]), y=r,r+1,...,[theta], where r is a known positive integer, and [theta][set membership, variant][Theta]={r,r+1,...} is an unknown parameter. We show that, for estimating [theta], 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 [theta] of a finite population whose elements are labeled from 1 to [theta], 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 [theta] 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).