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Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis

Author

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  • James E. Smith

    (Duke University, Durham, North Carolina)

Abstract

In many decision analysis problems, we have only limited information about the relevant probability distributions. In problems like these, it is natural to ask what conclusions can be drawn on the basis of this limited information. For example, in the early stages of analysis of a complex problem, we may have only limited fractile information for the distributions in the problem; what can we say about the optimal strategy or certainty equivalents given these few fractiles? This paper describes a very general framework for analyzing these kinds of problems where, given certain “moments” of a distribution, we can compute bounds on the expected value of an arbitrary “objective” function. By suitable choice of moment and objective functions we can formulate and solve many practical decision analysis problems. We describe the general framework and theoretical results, discuss computational strategies, and provide specific results for examples in dynamic programming, decision analysis with incomplete information, Bayesian statistics, and option pricing.

Suggested Citation

  • James E. Smith, 1995. "Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis," Operations Research, INFORMS, vol. 43(5), pages 807-825, October.
    • Handle: RePEc:inm:oropre:v:43:y:1995:i:5:p:807-825
    DOI: 10.1287/opre.43.5.807
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    Cited by:

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    4. J. A. Primbs, 2010. "SDP Relaxation of Arbitrage Pricing Bounds Based on Option Prices and Moments," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 137-155, January.
    5. Li, Xiaobo & Natarajan, Karthik & Teo, Chung-Piaw & Zheng, Zhichao, 2014. "Distributionally robust mixed integer linear programs: Persistency models with applications," European Journal of Operational Research, Elsevier, vol. 233(3), pages 459-473.
    6. 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.
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    9. Viet Anh Nguyen & Fan Zhang & Shanshan Wang & Jose Blanchet & Erick Delage & Yinyu Ye, 2021. "Robustifying Conditional Portfolio Decisions via Optimal Transport," Papers 2103.16451, arXiv.org, revised Apr 2024.
    10. Ran, Cuiling & Zhang, Yanzi & Yin, Ying, 2021. "Demand response to improve the shared electric vehicle planning: Managerial insights, sustainable benefits," Applied Energy, Elsevier, vol. 292(C).
    11. Dimitris Bertsimas & Ioana Popescu, 2002. "On the Relation Between Option and Stock Prices: A Convex Optimization Approach," Operations Research, INFORMS, vol. 50(2), pages 358-374, April.
    12. Carrasco, Vinicius & Farinha Luz, Vitor & Kos, Nenad & Messner, Matthias & Monteiro, Paulo & Moreira, Humberto, 2018. "Optimal selling mechanisms under moment conditions," Journal of Economic Theory, Elsevier, vol. 177(C), pages 245-279.
    13. Li, Zhaolin, 2021. "Robust Moral Hazard with Distributional Ambiguity," Working Papers BAWP-2021-01, University of Sydney Business School, Discipline of Business Analytics.
    14. J. Eric Bickel & James E. Smith, 2006. "Optimal Sequential Exploration: A Binary Learning Model," Decision Analysis, INFORMS, vol. 3(1), pages 16-32, March.
    15. Bernhard Kasberger, 2022. "An Equilibrium Model of the First-Price Auction with Strategic Uncertainty: Theory and Empirics," Papers 2202.07517, arXiv.org, revised Mar 2022.
    16. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.
    17. Villegas, Andrés M. & Medaglia, Andrés L. & Zuluaga, Luis F., 2012. "Computing bounds on the expected payoff of Alternative Risk Transfer products," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 271-281.
    18. John D. Rice & Brent A. Johnson & Robert L. Strawderman, 2022. "Screening for chronic diseases: optimizing lead time through balancing prescribed frequency and individual adherence," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(4), pages 605-636, October.
    19. Derek Singh & Shuzhong Zhang, 2020. "Tight Bounds for a Class of Data-Driven Distributionally Robust Risk Measures," Papers 2010.05398, arXiv.org, revised Oct 2020.
    20. Zuluaga, Luis F. & Peña, Javier & Du, Donglei, 2009. "Third-order extensions of Lo's semiparametric bound for European call options," European Journal of Operational Research, Elsevier, vol. 198(2), pages 557-570, October.
    21. Aleksandrina Goeva & Henry Lam & Huajie Qian & Bo Zhang, 2019. "Optimization-Based Calibration of Simulation Input Models," Operations Research, INFORMS, vol. 67(5), pages 1362-1382, September.
    22. Ioana Popescu, 2007. "Robust Mean-Covariance Solutions for Stochastic Optimization," Operations Research, INFORMS, vol. 55(1), pages 98-112, February.
    23. Henry Lam & Clementine Mottet, 2017. "Tail Analysis Without Parametric Models: A Worst-Case Perspective," Operations Research, INFORMS, vol. 65(6), pages 1696-1711, December.
    24. Zhaolin Li & Samuel N. Kirshner, 2021. "Salesforce Compensation and Two‐Sided Ambiguity: Robust Moral Hazard with Moment Information," Production and Operations Management, Production and Operations Management Society, vol. 30(9), pages 2944-2961, September.
    25. Pandit, Charuhas & Meyn, Sean, 2006. "Worst-case large-deviation asymptotics with application to queueing and information theory," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 724-756, May.

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