Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T09:42:09.641Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Special processes Numerical approximation and computational geometry Stochastic processes

Published online by Cambridge University Press:  03 December 2020

Mads Stehr  and
Markus Kiderlen
Mads Stehr*
Affiliation:
Aarhus University
Markus Kiderlen*
Affiliation:
Aarhus University
*
*Postal address: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark.
*Postal address: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Point processstationary stochastic processrandomized Newton–Cotes quadraturenumerical integrationasymptotic variance boundsrenewal process

MSC classification

Primary: 65D30: Numerical integration 60G55: Point processes
Secondary: 60G10: Stationary processes 60K05: Renewal theory 65D32: Quadrature and cubature formulas
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1284 - 1307
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (2003). Applied Probability and Queues (Stochastic Modelling and Applied Probability). Springer, New York.Google Scholar
Baddeley, A., Dorph-Petersen, K.-A. and Jensen, E. B. V. (2006). A note on the stereological implications of irregular spacing of sections. J. Microscopy 222, 177–181.CrossRefGoogle ScholarPubMed
Baddeley, A. and Jensen, E. B. V. (2004). Stereology for Statisticians (Chapman & Hall/CRC Monographs on Statistics & Applied Probability). CRC Press, Boca Raton.CrossRefGoogle Scholar
De Barra, G. (1981). Measure Theory and Integration. New Age International, New Delhi.Google Scholar
Doukhan, P. (1994). Mixing: Properties and Examples (Lecture Notes Statist.). Springer, New York.Google Scholar
Kiderlen, M. and Dorph-Petersen, K.-A. (2017). The Cavalieri estimator with unequal section spacing revisited. Image Anal. Stereol. 36, 135–141.10.5566/ias.1723CrossRefGoogle Scholar
Kiêu, K. (1997). Three lectures on systematic geometric sampling. Tech. Rep., Aarhus University.Google Scholar
Knopp, K. (1947). Theorie und Anwendung der unendlichen Reihen (Grundlehren der mathematischen Wissenschaften). Springer, Berlin.CrossRefGoogle Scholar
Matheron, G. (1965). Les variables régionalisées et leur estimation. Masson, Paris.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Heidelberg.CrossRefGoogle Scholar
Stehr, M. Stereology and spatio-temporal models. Doctoral Thesis, Aarhus University. Available at https://pure.au.dk/portal/da/publications/stereology-and-spatiotemporal-models(684bb552-d06e-4fb2-9e4a-3998b35e81e9).html.Google Scholar
Stehr, M. and Kiderlen, M. (2020). Supplementary material: asymptotic variance of Newton–Cotes quadratures based on randomized sampling points. Adv. Appl. Prob. Available at https://doi.org/10.1017/apr.2020.41.CrossRefGoogle Scholar
Stoer, J. and Bulirsch, R. (1993). Introduction to Numerical Analysis (Texts in Applied Mathematics). Springer, New York.CrossRefGoogle Scholar
Ziegel, J., Baddeley, A., Dorph-Petersen, K.-A. and Jensen, E. B. V. (2010). Systematic sampling with errors in sample locations. Biometrika 97, 1–13.CrossRefGoogle Scholar
Supplementary material: File

Stehr and Kiderlen supplementary material

Stehr and Kiderlen supplementary material

Download Stehr and Kiderlen supplementary material(File)
File 321.7 KB