On the estimation of the size of a finite population

We construct some estimates of the unknown size $N$ of finite population which are based on the sample of size $n$ drawn with replacement from this population. For the case when $N$, $n\to\infty$ and $0<\alpha_1\le \alpha=\frac{n}{N}\le\alpha_2<\infty$ (where $\alpha_1$ and $\alpha_2$ are given constants) a class of consistent uniformly asymptotically normal estimates of the parameter $\alpha$ is introduced. An asymptotically optimal (in this class) estimate is shown to be a function of the number $\eta_n$ of different elements in the sample.