On the inclusion probabilities in some unequal probability sampling plans without replacement

TL;DR

This paper compares inclusion probabilities in unequal probability sampling plans without replacement, showing how they become more uniform with larger sample sizes and differ between sampling methods, confirming a conjecture of He1jek.

Contribution

It provides new comparison results for inclusion probabilities in successive and rejective sampling, confirming a conjecture and analyzing their uniformity and divergence.

Findings

Inclusion probabilities become more uniform with larger sample sizes.

Rejective sampling yields more uniform inclusion probabilities than successive sampling.

Successive sampling's inclusion probabilities are more proportional to drawing probabilities.

Abstract

Comparison results are obtained for the inclusion probabilities in some unequal probability sampling plans without replacement. For either successive sampling or H\'{a}jek's rejective sampling, the larger the sample size, the more uniform the inclusion probabilities in the sense of majorization. In particular, the inclusion probabilities are more uniform than the drawing probabilities. For the same sample size, and given the same set of drawing probabilities, the inclusion probabilities are more uniform for rejective sampling than for successive sampling. This last result confirms a conjecture of H\'{a}jek (Sampling from a Finite Population (1981) Dekker). Results are also presented in terms of the Kullback--Leibler divergence, showing that the inclusion probabilities for successive sampling are more proportional to the drawing probabilities.