Asymptotic Analysis of Minimax Strategies in Survey Sampling

Abstract

Suppose that real numbers $y_i$ are associated with the units $i = 1, 2, \ldots, N$ of a population $U$ and that the vector $y = (y_1, y_2, \ldots, y_N)$ is known to be an element of the parameter space $\Theta$. The statistician has to select a sample $s \subset U$ of $n$ units and to employ $y_i, i \in s,$ to estimate $\bar{y} = \sum y_i/N.$ We propose to base this decision on an asymptotic version of the minimax principle. The asymptotically minimax principle is applied to three parameter spaces, including the parameter space considered by Scott and Smith and a space discussed by Cheng and Li. It turns out that stratified sampling is asymptotically minimax if the allocation is adapted to the parameter space. In addition we show that the commonly used ratio strategy [i.e., simple random sampling (srs) together with ratio estimation] and the RHC-strategy (see Rao, Hartley and Cochran) are asymptotically minimax with respect to parameter spaces chosen appropriately.