A Minimax Approach to Sample Surveys

Abstract

Suppose that there is a population of $N$ identifiable units each with two associated values $x_i$ and $y_i$. All $N$ values of $x$ are given but $y_i$ is determined only after the $i$th unit is selected and observed. The objective is to estimate the population total $\sum^N_{i=1} y_i$. It is assumed that $y_i = \theta x_i + \delta_ig(x_i), 1 \leq i \leq N$, where $(\delta_1, \cdots, \delta_N)$ is in some bounded neighborhood of $(0, \cdots 0)$. The Rao-Hartley-Cochran and Hansen-Hurwitz strategies are shown to be approximately minimax under certain models with $g(x) = x^{1/2}$ and with $g(x) = x$, the latter relating to a problem considered by Scott and Smith (1975). These two strategies are then compared with some commonly-used strategies and are found to perform favorably when $g^2(x)/x$ is an increasing function of $x$. The problem of estimating $\theta$ is also considered. Finally, some exact minimax results are obtained for sample size one.