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Accession Number ADA580236
Title Random Variables, Monotone Relations and Convex Analysis.
Publication Date Dec 2012
Media Count 33p
Personal Author J. O. Royset R. T. Rockafellar
Abstract 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.
Keywords Comonotonicity
Conjugate duality
Convergence in distribution
Convex analysis
Distribution functions
Measures of risk
Monotone functions
Random variables
Stochastic dominance
Stochastic optimization

Source Agency Non Paid ADAS
NTIS Subject Category 72F - Statistical Analysis
Corporate Author Naval Postgraduate School, Monterey, CA. Dept. of Operations Research.
Document Type Technical report
Title Note N/A
NTIS Issue Number 1325
Contract Number FA9550-11-1-0206 F1ATAO1194GOO1

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