Weighted finite population sampling to maximize entropy. (English) Zbl 0816.62008

Chen, Xiang-Hui; Dempster, Arthur P.; Liu, Jun S.

Biometrika 81, No. 3, 457-469 (1994).

Summary: 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 H. Stern and T. M. Cover [J. Am. Stat. Assoc. 84, 980-985 (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’ \(w_ i\). We show the essential uniqueness of the \(w_ i\) given the \(\pi_ i\), and describe practical, geometrically convergent algorithms for computing the \(w_ i\) from 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 \(0 < \pi_{ij} < \pi_ i\pi_ j\), which guarantees the Yates and Grundy variance estimator to be unbiased, definable for all samples and always nonnegative for any sample size.

MSC:

62D05	Sampling theory, sample surveys
62B10	Statistical aspects of information-theoretic topics

Keywords:

second-order inclusion probabilities; exponential family; independent Bernoulli trials; iterative proportional fitting; maximum entropy; rotatability; sampling with unequal probabilities without replacement; weighted sampling; coverage probabilities; geometrically convergent algorithms; Yates and Grundy variance estimator