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Tight tail probability bounds for distribution-free decision making

Author

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  • Roos, Ernst
  • Brekelmans, Ruud
  • van Eekelen, Wouter
  • den Hertog, Dick
  • van Leeuwaarden, Johan S.H.

Abstract

Chebyshev’s inequality provides an upper bound on the tail probability of a random variable based on its mean and variance. While tight, the inequality has been criticized for only being attained by pathological distributions that abuse the unboundedness of the underlying support and are not considered realistic in many applications. We provide alternative tight lower and upper bounds on the tail probability given a bounded support, mean and mean absolute deviation of the random variable. We obtain these bounds as exact solutions to semi-infinite linear programs. We apply the bounds for distribution-free analysis of the newsvendor model, stop-loss reinsurance and a problem from radiotherapy optimization with an ambiguous chance constraint.

Suggested Citation

  • Roos, Ernst & Brekelmans, Ruud & van Eekelen, Wouter & den Hertog, Dick & van Leeuwaarden, Johan S.H., 2022. "Tight tail probability bounds for distribution-free decision making," European Journal of Operational Research, Elsevier, vol. 299(3), pages 931-944.
    • Handle: RePEc:eee:ejores:v:299:y:2022:i:3:p:931-944
    DOI: 10.1016/j.ejor.2021.12.010
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    References listed on IDEAS

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    1. Heijnen, B., 1990. "Best upper and lower bounds on modified stop loss premiums in case of known range, mode, mean and variance of the original risk," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 207-220, September.
    2. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, June.
    3. Jansen, K. & Haezendonck, J. & Goovaerts, M. J., 1986. "Upper bounds on stop-loss premiums in case of known moments up to the fourth order," Insurance: Mathematics and Economics, Elsevier, vol. 5(4), pages 315-334, October.
    4. Karthik Natarajan & Melvyn Sim & Joline Uichanco, 2018. "Asymmetry and Ambiguity in Newsvendor Models," Management Science, INFORMS, vol. 64(7), pages 3146-3167, July.
    5. Shubhechyya Ghosal & Wolfram Wiesemann, 2020. "The Distributionally Robust Chance-Constrained Vehicle Routing Problem," Operations Research, INFORMS, vol. 68(3), pages 716-732, May.
    6. Aharon Ben-Tal & Eithan Hochman, 1976. "Stochastic Programs with Incomplete Information," Operations Research, INFORMS, vol. 24(2), pages 336-347, April.
    7. Georgia Perakis & Guillaume Roels, 2008. "Regret in the Newsvendor Model with Partial Information," Operations Research, INFORMS, vol. 56(1), pages 188-203, February.
    8. Napat Rujeerapaiboon & Daniel Kuhn & Wolfram Wiesemann, 2018. "Chebyshev Inequalities for Products of Random Variables," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 887-918, August.
    9. Jinfeng Yue & Bintong Chen & Min-Chiang Wang, 2006. "Expected Value of Distribution Information for the Newsvendor Problem," Operations Research, INFORMS, vol. 54(6), pages 1128-1136, December.
    10. De Vylder, F. & Goovaerts, M. J., 1982. "Analytical best upper bounds on stop-loss premiums," Insurance: Mathematics and Economics, Elsevier, vol. 1(3), pages 163-175, July.
    11. Alexander Shapiro & Jos Berge, 2002. "Statistical inference of minimum rank factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 67(1), pages 79-94, March.
    12. Ioana Popescu, 2005. "A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 632-657, August.
    13. Sungyong Choi & Andrzej Ruszczyński & Yao Zhao, 2011. "A Multiproduct Risk-Averse Newsvendor with Law-Invariant Coherent Measures of Risk," Operations Research, INFORMS, vol. 59(2), pages 346-364, April.
    14. Xin, Linwei & Goldberg, David A., 2021. "Time (in)consistency of multistage distributionally robust inventory models with moment constraints," European Journal of Operational Research, Elsevier, vol. 289(3), pages 1127-1141.
    15. Grani A. Hanasusanto & Vladimir Roitch & Daniel Kuhn & Wolfram Wiesemann, 2017. "Ambiguous Joint Chance Constraints Under Mean and Dispersion Information," Operations Research, INFORMS, vol. 65(3), pages 751-767, June.
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