Weighted finite population sampling to maximize entropy.

*(English)*Zbl 0816.62008Summary: 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 |