Abstract
In this paper we study approximations for the boundary crossing probabilities of moving sums of i.i.d. normal random variables. We approximate a discrete time problem with a continuous time problem allowing us to apply established theory for stationary Gaussian processes. By then subsequently correcting approximations for discrete time, we show that the developed approximations are very accurate even for a small window length. Also, they have high accuracy when the original r.v. are not exactly normal and when the weights in the moving window are not all equal. We then provide accurate and simple approximations for ARL, the average run length until crossing the boundary.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bauer P, Hackl P (1980) An extension of the MOSUM technique for quality control. Technometrics 22(1):1–7
Chu CSJ, Hornik K, Kaun CM (1995) MOSUM Tests for parameter constancy. Biometrika 82(3):603–617
Eiauer P, Hackl P (1978) The use of MOSUMS for quality control. Technometrics 20(4):431–436
Genz A, Bretz F (2009) Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics. Springer, Heidelberg
Genz A, Bretz F, Miwa T, Mi X, Leisch F, Scheipl F, Hothorn T (2018) mvtnorm: Multivariate Normal and t Distributions. https://CRAN.R-project.org/package=mvtnorm. R package version 1.0-8
Glaz J, Johnson B (1988) Boundary crossing for moving sums. J Appl Probab 25(1):81–88
Glaz J, Naus J, Wang X (2012) Approximations and inequalities for moving sums. Methodol Comput Appl Probab 14(3):597–616
Glaz J, Naus JI (1991) Tight bounds and approximations for scan statistic probabilities for discrete data. Ann Appl Probab 1(2):306–318
Glaz J, Naus JI, Wallenstein S, Wallenstein S, Naus JI (2001) Scan Statistics. Springer
Glaz J, Pozdnyakov V, Wallenstein S (2009) Scan Statistics: Methods and Applications. Birkhäuser, Boston
Haiman G (1999) First passage time for some stationary processes. Stoch Process Appl 80(2):231–248
Mohamed J, Delves L (1985) Computational methods for integral equations. Cambridge University Press
Moskvina V, Zhigljavsky A (2003) An algorithm based on Singular Spectrum Analysis for change-point detection. Commun Stat Simul Comput 32(2):319–352
Noonan J, Zhigljavsky A (2019) Approximations for the boundary crossing probabilities of moving sums of normal random variables. Communications in Statistics-Simulation and Computation, 1–22
Reed M, Simon B (1979) Methods of Modern Mathematical Physics: Scattering theory. Vol. 3 Academic Press
Shepp L (1971) First passage time for a particular Gaussian process. Ann Math Stat 42(3):946–951
Siegmund D (1985) Sequential analysis: Tests and confidence intervals. Springer Science & Business Media
Siegmund D (1986) Boundary crossing probabilities and statistical applications. Ann Stat 14(2):361–404
Slepian D (1961) First passage time for a particular Gaussian process. Ann Math Stat 32(2):610–612
Waldmann KH (1986) Bounds to the distribution of the run length in general quality-control schemes. Statistische Hefte 27(1):37
Wang X, Glaz J (2014) Variable window scan statistics for normal data. Commun Stat Theory Meth 43(10-12):2489–2504
Wang X, Zhao B, Glaz J (2014) A multiple window scan statistic for time series models. Stat Probab Lett 94:196–203
Xia Z, Guo P, Zhao W (2009) Monitoring structural changes in generalized linear models. Commun Stat Theory Meth 38(11):1927–1947
Acknowledgment
The authors are grateful to the referees for careful reading of the manuscript and useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Noonan, J., Zhigljavsky, A. Approximations for the Boundary Crossing Probabilities of Moving Sums of Random Variables. Methodol Comput Appl Probab 23, 873–892 (2021). https://doi.org/10.1007/s11009-019-09769-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-019-09769-7