A Model for Studying Robustness of Estimators and Informativeness of Labels in Sampling with Varying Probabilities

Summary This paper presents a model for investigating how information contained in the labels of a finite population may be utilized in order to reduce variance when sampling from a finite population with varying probabilities. The model assumes a population of N values of a characteristic X given as Xi = Yi + Zi (i = 1, …, N), where the mean of X is to be estimated. Here Yi is a systematic component in the sense that its relation to the label i is a priori known, or intelligently guessed at, while Zi is an error component. In the with-replacement case, a sample of labels is selected with possibly unequal probabilities pi,(i = 1, …, N) attached to the various labels in each of the n draws. Assuming that N is reasonably large, the variance of the usual estimator is studied under various assumptions about the shape of the continuous distribution approximating the Y-distribution (e.g. gamma, Pareto, Weibull shapes), various assumptions about the influence of the error component Z, and different designs for distributing the probabilities Pi among the N labels.