On a characterization of ordered pivotal sampling

TL;DR

This paper characterizes pivotal sampling within the cube method framework, demonstrating its equivalence to systematic sampling under fixed ordering and enabling variance reduction through unit ordering.

Contribution

It provides a novel characterization of pivotal sampling, allowing for the computation of second-order inclusion probabilities and insights into variance reduction strategies.

Findings

Pivotal sampling is equivalent to systematic sampling with fixed order.

The characterization allows calculation of second-order inclusion probabilities.

Ordering units can reduce variance without significant efficiency loss.

Abstract

When auxiliary information is available at the design stage, samples may be selected by means of balanced sampling. Deville and Tille proposed in 2004 a general algorithm to perform balanced sampling, named the cube method. In this paper, we are interested in a particular case of the cube method named pivotal sampling, and first described by Deville and Tille in 1998. We show that this sampling algorithm, when applied to units ranked in a fixed order, is equivalent to Deville's systematic sampling, in the sense that both algorithms lead to the same sampling design. This characterization enables the computation of the second-order inclusion probabilities for pivotal sampling. We show that the pivotal sampling enables to take account of an appropriate ordering of the units to achieve a variance reduction, while limiting the loss of efficiency if the ordering is not appropriate.