Convex Sets of Finite Population Plans

Abstract

Let $P_1$ be a finite population sampling plan and $V$ a collection of subsets of units. The inclusion probabilities for members of $V$ may be calculated. For example, if $V$ comprises all single units and pairs of units we obtain all first and second order inclusion probabilities $\pi_i, \pi_{ij}$. Another plan $P_2$ is called equivalent to $P_1$ with respect to $V$ if the corresponding inclusion probabilities for $P_1$ are equal to those for $P_2$. However, $P_2$ may have fewer samples with positive probability of selection, that is to say smaller "support." An upper bound is put on the minimum support size of all such $P_2$. For $P_1$ simple random sampling, some examples are given for $P_2$ with small support.