Exactness and Ordering in Discrete Cavalieri Sampling

Abstract

We study the discrete Cavalieri estimator under systematic sampling from a finite population, which models an object represented by a finite sequence of blocks along a sampling axis.
For a fixed population size and a sample size that divides it, we characterize when the estimator has zero variance, namely exactness, through an explicit balance condition that characterizes the zero-variance populations; this turns out to be a simple linear family.
We then ask when exactness continues to hold if the sample size is allowed to vary within the even divisors of an even population size.
In that case, we prove that exactness across all such even sample sizes necessarily implies the matched-pairs condition that is known to be sufficient at a fixed even sample size.
We also derive a variance formula showing that it depends only on how much the sums over certain groups differ from their average.
This leads to a concrete partitioning objective for choosing an ordering and helps explain why exact optimization quickly becomes impractical.
Guided by this objective and by smooth fractionator practice, we discuss simple heuristics and show that a pairing-based ordering is exact under a simple affine model and remains stable under bounded perturbations.