Improved variance prediction for systematic sampling on ℝ

• Published in: Statistics (Berlin, DDR) DDR), 2004-06, Vol.38 (3), p.243-272
• Main Authors: García-Fiñana, Marta, Cruz-Orive, Luis M.
• Format: Article
• Language: English
• Subjects: Cavalieri sampling; Euler-MacLaurin summation formula; Extension term; Fourier transform; Fractional derivatives; Matheron's transitive approach; q-smoothness; Variance estimation
• ISSN: 0233-1888; 1029-4910
• DOI: 10.1080/0233188032000158826

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 Qˆ 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(Qˆ) in terms of the smoothness properties of f. We extend the current theory, which holds for smoothness constant q ∈ ℕ, to any q ≥ 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 ≥ 0; we concentrate on the useful case q ∈ [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. * Current address: Centre for Medical Statistics and Health Evaluation, University of Liverpool, Brownlow Street, Liverpool L69 3GS, UK