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**The construction of \(\Pi\) PS sampling designs through a method of emptying boxes.**
*(English)*
Zbl 0701.62019

Let \(X_ 1,X_ 2,...,X_ n\) be known measures of size for the units \(U_ 1,U_ 2,...,U_ N\) of a finite population. It is required to select a sample of size n such that the probability, \(\pi_ i\), of inclusion of the unit \(U_ i\) in the sample is proportional to its size, i.e. \(\pi_ i=nX_ i/\sum^{N}_{i=1}X_ i.\) Many such \(\Pi\) PS sampling methods have been suggested in the literature with varying degrees of complexity.

The authors present a simple, computer-adaptable method. It consists of 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 for emptying out all the boxes by this method are derived. It is then shown that every \(\Pi\) PS design can be constructed from such a game. Also several other properties of the design are shown to be satisfied.

The authors present a simple, computer-adaptable method. It consists of 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 for emptying out all the boxes by this method are derived. It is then shown that every \(\Pi\) PS design can be constructed from such a game. Also several other properties of the design are shown to be satisfied.

Reviewer: T.J.Rao