# Weighted Finite Population Sampling to Maximize Entropy

Xiang-Hui Chen, Arthur P. Dempster and Jun S. Liu
Biometrika
Vol. 81, No. 3 (Aug., 1994), pp. 457-469
DOI: 10.2307/2337119
Stable URL: http://www.jstor.org/stable/2337119
Page Count: 13

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## Abstract

Attention is drawn to a method of sampling a finite population of N units with unequal probabilities and without replacement. The method was originally proposed by Stern & Cover (1989) as a model for lotteries. The method can be characterized as maximizing entropy given coverage probabilities πi, or equivalently as having the probability of a selected sample proportional to the product of a set of `weights' wi. We show the essential uniqueness of the wi given the πi, and describe practical, geometrically convergent algorithms for computing the wi from the πi. We present two methods for stepwise selection of sampling units, and corresponding schemes for removal of units that can be used in connection with sample rotation. Inclusion probabilities of any order can be written explicitly in closed form. Second-order inclusion probabilities πij satisfy the condition $0 < \pi_{ij} < \pi_i \pi_j$, which guarantees Yates & Grundy's variance estimator to be unbiased, definable for all samples and always nonnegative for any sample size.

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