Random variables, monotone relations, and convex analysis

• 成果类型: Article
• 署名作者: Rockafellar, R. T.; Royset, J. O.
• 署名单位: University of Washington; University of Washington Seattle; United States Department of Defense; United States Navy; Naval Postgraduate School
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0801-1
• 发表日期: 2014
• 页码: 297-331
• 关键词: MEAN-RISK MODELS; Value-at-risk; stochastic-dominance; regression; optimization

摘要

Random variables can be described by their cumulative distribution functions, a class of nondecreasing functions on the real line. Those functions can in turn be identified, after the possible vertical gaps in their graphs are filled in, with maximal monotone relations. Such relations are known to be the subdifferentials of convex functions. Analysis of these connections yields new insights. The generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations. In subdifferential terms, it corresponds to passing to conjugate convex functions under the Legendre-Fenchel transform. Among other things, this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions. Measures of risk that employ quantiles (VaR) and superquantiles (CVaR), either individually or in mixtures, are illuminated in this way. Formulas for their calculation are seen from a perspective that reveals how they were discovered. The approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying superdistribution. In this way a generalization of Koenker-Basset error is derived which lays a foundation for superquantile regression as a higher-order extension of quantile regression.