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Improved variance prediction for systematic sampling on \(\mathbb{R}\). (English) Zbl 1063.62009

Summary: 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 \(\mathbb{R}\). The unbiased estimator \(\widehat Q\) 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\((\widehat Q)\) in terms of the smoothness properties of \(f\). We extend the current theory, which holds for the smoothness constant \(q\in \mathbb{N}\), to any \(q\geq 0\); 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 \(q\geq 0\); we concentrate on the useful case \(q\in[0,1]\). By means of synthetic data, and real data from a human brain, we show that the new estimator performs better than its current alternatives.

MSC:

62D05 Sampling theory, sample surveys
65B15 Euler-Maclaurin formula in numerical analysis
65D30 Numerical integration
26A33 Fractional derivatives and integrals
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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