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Asymptotic variance of Newton-Cotes quadratures based on randomized sampling points

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Asymptotic variance of Newton-Cotes quadratures based on randomized sampling points. / Stehr, Mads; Kiderlen, Markus.

In: Advances in Applied Probability, Vol. 52, No. 4, 2020, p. 1284-1307.

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Stehr, Mads ; Kiderlen, Markus. / Asymptotic variance of Newton-Cotes quadratures based on randomized sampling points. In: Advances in Applied Probability. 2020 ; Vol. 52, No. 4. pp. 1284-1307.

Bibtex

@article{c9a32c1a72fb42379268f2eedd0aa4ea,
title = "Asymptotic variance of Newton-Cotes quadratures based on randomized sampling points",
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 naive 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.",
keywords = "Point process, stationary stochastic process, randomized Newton-Cotes quadrature, numerical integration, asymptotic variance bounds, renewal process, Point process, stationary stochastic process, randomized Newton-Cotes quadrature, numerical integration, asymptotic variance bounds, renewal process",
author = "Mads Stehr and Markus Kiderlen",
year = "2020",
doi = "10.1017/apr.2020.41",
language = "English",
volume = "52",
pages = "1284--1307",
journal = "Advances in Applied Probability",
issn = "0001-8678",
publisher = "Applied Probability Trust",
number = "4",

}

RIS

TY - JOUR

T1 - Asymptotic variance of Newton-Cotes quadratures based on randomized sampling points

AU - Stehr, Mads

AU - Kiderlen, Markus

PY - 2020

Y1 - 2020

N2 - 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 naive 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.

AB - 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 naive 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.

KW - Point process

KW - stationary stochastic process

KW - randomized Newton-Cotes quadrature

KW - numerical integration

KW - asymptotic variance bounds

KW - renewal process

KW - Point process

KW - stationary stochastic process

KW - randomized Newton-Cotes quadrature

KW - numerical integration

KW - asymptotic variance bounds

KW - renewal process

U2 - 10.1017/apr.2020.41

DO - 10.1017/apr.2020.41

M3 - Journal article

VL - 52

SP - 1284

EP - 1307

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 4

ER -