## The Annals of Statistics

### Asymptotic Analysis of Minimax Strategies in Survey Sampling

Horst Stenger

#### 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.

#### Article information

Source
Ann. Statist., Volume 17, Number 3 (1989), 1301-1314.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347270

Digital Object Identifier
doi:10.1214/aos/1176347270

Mathematical Reviews number (MathSciNet)
MR1015152

Zentralblatt MATH identifier
0719.62022

JSTOR
links.jstor.org

Subjects
Primary: 62D05: Sampling theory, sample surveys
Secondary: 62C20: Minimax procedures

#### Citation

Stenger, Horst. Asymptotic Analysis of Minimax Strategies in Survey Sampling. Ann. Statist. 17 (1989), no. 3, 1301--1314. doi:10.1214/aos/1176347270. https://projecteuclid.org/euclid.aos/1176347270