## The Annals of Statistics

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

### 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.

#### 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

**Keywords**

Asymptotic analysis invariance minimax strategies Rao-Hartley-Cochran strategy ratio strategy sample surveys

#### 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