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Biometrika 1994 81(3):457-469; doi:10.1093/biomet/81.3.457
© 1994 by Biometrika Trust
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Weighted finite population sampling to maximize entropy

XIANG-HUI CHEN, ARTHUR P. DEMPSTER and JUN S. LIU

Department of Statistics, Harvard University Cambridge, Massachusetts 02138, U.S.A.

Received for publication 1 June 1993. Revision received 1 April 1994.
  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 {pi}i, 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 {pi}i. 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 {pi}ij satisfy the condition O < {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.

Key Words: Exponential family • Independent Bernoulli trials • Iterative proportional fitting • Maximum entropy • Rotatability • Sampling with unequal probabilities and without replacement • Survey sampling • Weighted sampling


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A Jackknife Variance Estimator for Unistage Stratified Samples with Unequal Probabilities
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