Asymptotic Variance of Newton–Cotes Quadratures Based on Randomized Sampling Points

Mads Stehr, Markus Kiderlen

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Abstract

We consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.
Original languageEnglish
JournalAdvances in Applied Probability
Volume52
Issue number4
Pages (from-to)1284-1307
Number of pages24
ISSN0001-8678
DOIs
Publication statusPublished - Dec 2020

Keywords

  • Point process
  • Stationary stochastic process
  • Randomized Newton-Cotes quadrature
  • Numerical integration
  • Asympotic variance bounds
  • Renewal process

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