Geodesic
On some problems of statistical inference for stratified populations
Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 113-118

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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. 
@article{TVP_1984_29_1_a9,
     author = {G. J. Iv\v{c}enko},
     title = {On some problems of statistical inference for stratified populations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {113--118},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a9/}
}
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G. J. Ivčenko. On some problems of statistical inference for stratified populations. Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 113-118. http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a9/