Random variables, monotone relations, and convex analysis.

*(English)*Zbl 1330.60009The authors investigate how different concepts of probability and statistics may be treated within the framework of convex analysis. The relation between distribution functions, defined in the real line, and the quantile functions on \((0, 1)\), are linked using results of convex functions.

It is of interest for statisticians the fact that quantile regression, a well-known alternative model for regression fitting, can be bootstrapped into a new higher/order approximation tool within the framework provided by superquantiles. For economists, the use of them in the study of conditional-value at-risk should be the main source of their interest. Is discussed how superquantiles obtained its importance coming from their role in stochastic optimization.

In the paper, the authors derive results on set convergence, maximal monotonicity from distributions and quantiles, and super expectation functions, among others. The results are reported in nine theorems and a corollary. Theorems 1 and 2 deal with super expectations; Theorem 3 and 7 and its corollary with superquantile functions. Convergence is characterized in Theorems 4 and 5; Theorem 8 deals with first-order stochastic dominance and Theorem 9 with characterizations of co-monotonicity.

It is of interest for statisticians the fact that quantile regression, a well-known alternative model for regression fitting, can be bootstrapped into a new higher/order approximation tool within the framework provided by superquantiles. For economists, the use of them in the study of conditional-value at-risk should be the main source of their interest. Is discussed how superquantiles obtained its importance coming from their role in stochastic optimization.

In the paper, the authors derive results on set convergence, maximal monotonicity from distributions and quantiles, and super expectation functions, among others. The results are reported in nine theorems and a corollary. Theorems 1 and 2 deal with super expectations; Theorem 3 and 7 and its corollary with superquantile functions. Convergence is characterized in Theorems 4 and 5; Theorem 8 deals with first-order stochastic dominance and Theorem 9 with characterizations of co-monotonicity.

Reviewer: Carlos Narciso Bouza Herrera (Habana)

##### MSC:

60A99 | Foundations of probability theory |

52A41 | Convex functions and convex programs in convex geometry |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

90C15 | Stochastic programming |

90C25 | Convex programming |

##### Keywords:

random variables; quantiles; superquantiles; super expectations; super distributions; convergence in distribution; stochastic dominance; co-monotonicity; measures of risk; value-at-risk; conditional-value-at-risk; convex analysis; conjugate duality; stochastic optimization
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\textit{R. T. Rockafellar} and \textit{J. O. Royset}, Math. Program. 148, No. 1--2 (B), 297--331 (2014; Zbl 1330.60009)

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