On a Characterization of ordered Pivotal Sampling

When auxiliary information is available at the design stage, samples may be selected bymeans of balanced sampling. Deville and Tillé proposed in 2004 a general algorithm toperform balanced sampling, named the cube method. In this paper, we are interested in aparticular case of the cube method named pivotal sampling, and first described by Devilleand Tillé in 1998. We show that this sampling algorithm, when applied to units ranked ina fixed order, is equivalent to Deville’s systematic sampling, in the sense that both algorithmslead to the same sampling design. This characterization enables the computationof the second-order inclusion probabilities for pivotal sampling. We show that the pivotalsampling enables to take account of an appropriate ordering of the units to achieve avariance reduction, while limiting the loss of efficiency if the ordering is not appropriate.