WEIGHTED FINITE POPULATION-SAMPLING TO MAXIMIZE ENTROPY

Attention is drawn to a method of sampling a finite population of N un its with unequal probabilities and without replacement. The method was originally proposed by Stern and Cover (1989) as a model for lotterie s. The method can be characterized as maximizing entropy given coverag e 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 de scribe practical, geometrically convergent algorithms for computing th e w(i) from the pi(i). We present two methods for stepwise-selection o f sampling units, and corresponding schemes for removal of units that can be used in connection with sample rotation. Inclusion probabilitie s 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 Yates and Grundy's variance estimator to be unbiased, definable for all samples and always nonnegative for any sam ple size.