Improved variance prediction for systematic sampling on R

Many problems, in stereology and elsewhere (geometric sampling, calculus, etc.) reduce to estimating the integral Q of a non-random measurement function f over a bounded support on R. The unbiased estimator 6 based on systematic sampling of period T > 0 (such as the popular Cavalieri estimator) is usually convenient and highly precise. The purpose of this paper is twofold. First, to obtain a new, general representation of var((Q) over cap) in terms of the smoothness properties of f. We extend the current theory, which holds for smoothness constant q is an element of N, to any q greater than or equal to 0; to this end we develop a new version of the Euler-MacLaurin summation formula, making use of fractional calculus. Our second purpose is to apply the mentioned representation to obtain a new variance estimator for any q greater than or equal to 0: we concentrate on the useful case q is an element of [0, 1]. By means of synthetic data, and real data from a human brain, we show that the new estimator performs better than its Current alternatives.