Improved variance prediction for systematic sampling on ℝ

Abstract

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 estimatorQ 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) 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.

On the Precision of Curve Length Estimation in the Plane

“…17b was q = 0. Theory establishes that, under certain conditions, the trend or 'extension term' of the variance of a Cavalieri estimator is of order O(T 2q+2 ), (Kiêu et al, 1999;García-Fiñana and Cruz-Orive, 2004). In the present study the expected trend of CE 2 { B(ω, x)} should therefore be approximately of O(E(I) −(2q+2) ) = O(E(I) −2 ).…”

“…12b is the standard Cavalieri estimator of it. Moreover, because the measurement function I Y ω (z) is integer valued it will exhibit jumps, and therefore its smoothness constant will be q = 0 (Kiêu et al, 1999;García-Fiñana and Cruz-Orive, 2004). In consequence, the σ 2 i are computed with q = 0.…”

Sampling Intensity With Fixed Precision When Estimating Volume of Human Brain Compartments

“…where q ∈ [0, 1] and Γ() and ζ () denote the gamma function and the Riemann Zeta function, respectively (see García-Fiñana and Cruz-Orive, 2004;Cruz-Orive, 2006, where a table with numerical values for α(q) is provided, and references therein). The smoothness constant q can be estimated from the data sample as described in Souchet (1995) and García-Fiñana and Cruz-Orive (2004).…”

“…2 in powers of T (e.g., Matheron, 1965;1971;Gundersen and Jensen, 1987;Cruz-Orive, 1989;Kellerer, 1989;Kiêu, 1997;Gual-Arnau and Cruz-Orive, 1998;Gundersen et al, 1999;García-Fiñana and Cruz-Orive, 2004). A general and currently accepted variance estimator reads as follows:…”

“…The smoothness constant q can be estimated from the data sample as described in Souchet (1995) and García-Fiñana and Cruz-Orive (2004).…”