On some problems of statistical inference for stratified populations

Let a simple random sample with replacement of size $n$ be drawn from a finite population $U$ of size $N$, stratified into $k$ strata $U_1,\dots,U_k$ of sizes $N_1,\dots,N_k$ respectively. Let $\mu_{jr}$ be the number of elements from the stratum $U_j$ which appeared $r$ times ($j=1,\dots,k$; $r=1,\dots,n$). It is shown that the vector $\eta=(\eta_1,\dots,\eta_k)$, where $\displaystyle\eta_j=\sum_{r=1}^n\mu_{jr}$, $j=1,\dots,k$ is a complete sufficient statistic for $\mathbf N=(N_1,\dots,N_k)$. Unbiased minimum variance estimates for a class of parametric functions $\tau(\mathbf N)$ are constructed.