Improving the Cavalieri Estimator under Non-Equidistant Sampling and Dropouts

Mads Stehr, Markus Kiderlen

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.
Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models. The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries.
Original languageEnglish
JournalImage Analysis and Stereology
Volume39
Issue number3
Pages (from-to)197-212
Number of pages16
ISSN1580-3139
DOIs
Publication statusPublished - 2020

Keywords

  • Cavalieri estimator
  • Dropouts
  • Newton-Cotes quadrature
  • Numerical integration with random nodes
  • Stationary point process
  • Variance estimation
  • Weakly (m,p)-piecewise smooth function

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