On a characterization of ordered pivotal sampling

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.