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<p>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月27日

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  • Mads Stehr,
  • Markus Kiderlen

摘要

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.