Unequal probability sampling from a finite population: A multicriteria approach

In this note we address the problem of determining selection probabilities for multipurpose surveys, when the aim is the simultaneous minimization of variances for each variable under study. A characterization of the set of Pareto-optimal designs is given for designs with replacement and also for a class of designs without replacement, namely, Poisson designs. As an application, we describe a problem encountered in Auditing, where both the fraction of misstatements, and the average amount of such misstatements are of interest.