CVaR distance between univariate probability distributions and approximation problems.

*(English)*Zbl 1391.62025Summary: The paper defines new distances between univariate probability distributions, based on the concept of the CVaR norm. We consider the problem of approximation of a discrete distribution by some other discrete distribution. The approximating distribution has a smaller number of atoms than the original one. Such problems, for instance, must be solved for generation of scenarios in stochastic programming. The quality of the approximation is evaluated with new distances suggested in this paper. We use CVaR constraints to assure that the approximating distribution has tail characteristics similar to the target distribution. The numerical algorithm is based on two main steps: (i) optimal placement of positions of atoms of the approximating distribution with fixed probabilities; (ii) optimization of probabilities with fixed positions of atoms. These two steps are iterated to find both optimal atom positions and probabilities. Numerical experiments show high efficiency of the proposed algorithms, solved
with convex and linear programming.

##### MSC:

62E17 | Approximations to statistical distributions (nonasymptotic) |

91B30 | Risk theory, insurance (MSC2010) |

91G70 | Statistical methods; risk measures |

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\textit{K. Pavlikov} and \textit{S. Uryasev}, Ann. Oper. Res. 262, No. 1, 67--88 (2018; Zbl 1391.62025)

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