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Combination setwise‐Bonferroni‐type bounds

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  • Timothy M. Costigan

Abstract

We consider three classes of lower bounds to P(c) = P (X1 ≤ c1,…, Xn ≤ c); Bonferroni‐type bounds, product‐type bounds and setwise bounds. Setwise probability inequalities are shown to be a compromise between product‐type and Bonferroni‐type probability inequalities. Bonferroni‐type inequalities always hold. Product‐type inequalities require positive dependence conditions, but are superior to the Bonferroni‐type and setwise bounds when these conditions are satisfied. Setwise inequalities require less stringent positive dependence bound conditions than the product‐type bounds. Neither setwise nor Bonferroni‐type bounds dominate the other. Optimized setwise bounds are developed. Results pertaining to the nesting of setwise bounds are obtained. Combination setwise‐Bonferroni‐type bounds are developed in which high dimensional setwise bounds are applied and second and third order Bonferroni‐type bounds are applied within each subvector of the setwise bounds. These new combination bounds, which are applicable for associated random variables, are shown to be superior to Bonferroni‐type and setwise bounds for moving averages and runs probabilities. Recently proposed upper bounds to P(c) are reviewed. The lower and upper bounds are tabulated for various classes of multivariate normal distributions with banded covariance matrices. The bounds are shown to be surprisingly accurate and are much easier to compute than the inclusion‐exclusion bounds. A strategy for employing the bounds is developed. © 1996 John Wiley & Sons, Inc.

Suggested Citation

  • Timothy M. Costigan, 1996. "Combination setwise‐Bonferroni‐type bounds," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(1), pages 59-77, February.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:1:p:59-77
    DOI: 10.1002/(SICI)1520-6750(199602)43:13.0.CO;2-M
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    References listed on IDEAS

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    1. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    2. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    3. Hoppe, Fred M., 1985. "Iterating bonferroni bounds," Statistics & Probability Letters, Elsevier, vol. 3(3), pages 121-125, June.
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    1. Serkan Eryilmaz & Cihangir Kan & Fatih Akici, 2009. "Consecutive k‐within‐m‐out‐of‐n:F system with exchangeable components," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 503-510, September.
    2. Eryilmaz, Serkan & Unlu, Kamil Demirberk, 2023. "A new generalized δ-shock model and its application to 1-out-of-(m+1):G cold standby system," Reliability Engineering and System Safety, Elsevier, vol. 234(C).
    3. József Bukszár & Gergely Mádi-Nagy & Tamás Szántai, 2012. "Computing bounds for the probability of the union of events by different methods," Annals of Operations Research, Springer, vol. 201(1), pages 63-81, December.

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