CVaR (superquantile) norm: Stochastic case

Alexander Mafusalov and Stan Uryasev

European Journal of Operational Research, 2016, vol. 249, issue 1, 200-208

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.

Keywords: CVaR norm; L-p norm; Superquantile; Risk quadrangle; Linear regression (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (10)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:ejores:v:249:y:2016:i:1:p:200-208

DOI: 10.1016/j.ejor.2015.09.058

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