The Construction of $\Pi PS$ Sampling Designs Through a Method of Emptying Boxes

Abstract

We present a simple but universal technique for the construction of $\pi PS$ sampling designs. A tool that is used in the construction consists of playing a game in which objects are removed from $N$ boxes, $n$ at a time, and at most one from each box at a time. Necessary and sufficient conditions on $N, n$ and the contents of the boxes are established such that all boxes can be emptied by this process. It is shown that every $\pi PS$ design can be derived from such a game. Sampling designs with additional properties are obtained through additional restrictions on emptying the boxes. Various rigorous methods are presented, complemented by numerous suggestions. The emphasis is on controlling sample selection probabilities and inequalities for the first- and second-order inclusion probabilities. The method is very adaptive to computer use.