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CVaR (superquantile) norm: Stochastic case

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  • Mafusalov, Alexander
  • Uryasev, Stan

Abstract

The concept of Conditional Value-at-Risk (CVaR) is used in various applications in uncertain environment. This paper introduces CVaR (superquantile) norm for a random variable, which is by definition CVaR of absolute value of this random variable. It is proved that CVaR norm is indeed a norm in the space of random variables. CVaR norm is defined in two variations: scaled and non-scaled. L-1 and L-infinity norms are limiting cases of the CVaR norm. In continuous case, scaled CVaR norm is a conditional expectation of the random variable. A similar representation of CVaR norm is valid for discrete random variables. Several properties for scaled and non-scaled CVaR norm, as a function of confidence level, were proved. Dual norm for CVaR norm is proved to be the maximum of L-1 and scaled L-infinity norms. CVaR norm, as a Measure of Error, is related to a Regular Risk Quadrangle. Trimmed L1-norm, which is a non-convex extension for CVaR norm, is introduced analogously to function L-p for p < 1. Linear regression problems were solved by minimizing CVaR norm of regression residuals.

Suggested Citation

  • Mafusalov, Alexander & Uryasev, Stan, 2016. "CVaR (superquantile) norm: Stochastic case," European Journal of Operational Research, Elsevier, vol. 249(1), pages 200-208.
    • Handle: RePEc:eee:ejores:v:249:y:2016:i:1:p:200-208
    DOI: 10.1016/j.ejor.2015.09.058
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    References listed on IDEAS

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    Cited by:

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    3. Kuzmenko Viktor & Salam Romel & Uryasev Stan, 2020. "Checkerboard copula defined by sums of random variables," Dependence Modeling, De Gruyter, vol. 8(1), pages 70-92, January.
    4. Ivanov Roman V., 2018. "On risk measuring in the variance-gamma model," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 23-33, January.
    5. Kuzmenko Viktor & Salam Romel & Uryasev Stan, 2020. "Checkerboard copula defined by sums of random variables," Dependence Modeling, De Gruyter, vol. 8(1), pages 70-92, January.
    6. Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.
    7. Ramponi, Federico Alessandro & Campi, Marco C., 2018. "Expected shortfall: Heuristics and certificates," European Journal of Operational Research, Elsevier, vol. 267(3), pages 1003-1013.
    8. Konstantin Pavlikov & Stan Uryasev, 2018. "CVaR distance between univariate probability distributions and approximation problems," Annals of Operations Research, Springer, vol. 262(1), pages 67-88, March.

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