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Approximations and Inequalities for Moving Sums

Author

Listed:
  • Joseph Glaz

    (University of Connecticut)

  • Joseph Naus

    (The State University of New Jersey)

  • Xiao Wang

    (University of Connecticut)

Abstract

In this article accurate approximations and inequalities are derived for the distribution, expected stopping time and variance of the stopping time associated with moving sums of independent and identically distributed continuous random variables. Numerical results for a scan statistic based on a sequence of moving sums are presented for a normal distribution model, for both known and unknown mean and variance. The new R algorithms for the multivariate normal and t distributions established by Genz et al. (2010) provide readily available numerical values of the bounds and approximations.

Suggested Citation

  • Joseph Glaz & Joseph Naus & Xiao Wang, 2012. "Approximations and Inequalities for Moving Sums," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 597-616, September.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-011-9251-x
    DOI: 10.1007/s11009-011-9251-x
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    References listed on IDEAS

    as
    1. Anders Westlund, 1984. "Sequential moving sums of squares of OLS residuals in parameter stability testing," Quality & Quantity: International Journal of Methodology, Springer, vol. 18(3), pages 261-273, May.
    2. Haiman, George, 1999. "First passage time for some stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 231-248, April.
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    Citations

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    Cited by:

    1. Jack Noonan & Anatoly Zhigljavsky, 2021. "Approximations for the Boundary Crossing Probabilities of Moving Sums of Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 873-892, September.
    2. Qianzhu Wu & Joseph Glaz, 2019. "Robust Scan Statistics for Detecting a Local Change in Population Mean for Normal Data," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 295-314, March.
    3. Bo Zhao & Joseph Glaz, 2016. "Scan Statistics for Detecting a Local Change in Variance for Normal Data with Known Variance," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 563-573, June.
    4. Anat Reiner-Benaim, 2016. "Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 717-745, September.
    5. Alexandru Amarioarei & Cristian Preda, 2020. "One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems," Mathematics, MDPI, vol. 8(4), pages 1-11, April.
    6. Wang, Xiao & Zhao, Bo & Glaz, Joseph, 2014. "A multiple window scan statistic for time series models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 196-203.
    7. Haiman, George & Preda, Cristian, 2013. "One dimensional scan statistics generated by some dependent stationary sequences," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1457-1463.
    8. Qianzhu Wu & Joseph Glaz, 2021. "Scan Statistics for Normal Data with Outliers," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 429-458, March.
    9. Zhao, Bo & Glaz, Joseph, 2016. "Scan statistics for detecting a local change in variance for normal data with unknown population variance," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 137-145.

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