Confidence intervals for Newton–Cotes quadratures based on stationary point processes

Mads Stehr () and Markus Kiderlen ()
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Mads Stehr: Copenhagen Business School
Markus Kiderlen: Aarhus University

Statistical Methods & Applications, 2025, vol. 34, issue 1, No 3, 39-67

Abstract: Abstract Motivated by the stereological application of volume estimation, this paper is concerned with numerical integration on the real line, employing function values at a finite set of randomly chosen points. The sampling points are modeled by a stationary point process, with the estimators being Newton–Cotes quadratures. Our comprehensive probabilistic analysis crucially extends existing results regarding the approximation error and variance, accommodating more general integrands and non-ergodic sampling processes. Notably, these findings are used to formulate novel asymptotic confidence intervals, a considerable challenge given the usual absence of limit distributions. To underscore the practicality of our approach, we apply it to a stereological simulation study. Specifically, we establish confidence intervals for the volume of a three-dimensional ellipsoid, based on section areas obtained from randomly positioned parallel planes.

Keywords: Confidence interval; Perturbed systematic sampling; Randomized Newton–Cotes quadrature; Cavalieri volume estimator; Sobolev-type space of BV-functions (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10260-024-00773-x

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