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 LegendreFenchel 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 KoenkerBasset error is derived which lays a foundation for superquantile regression as a higherorder extension of quantile regression.

Keywords

Comonotonicity Conditionalvalueatrisk Conjugate duality Convergence in distribution Convex analysis Distribution functions Graphs Measures of risk Monotone functions Quantiles Random variables Stochastic dominance Stochastic optimization Superdistributions Superexpectations Superquantiles Valueatrisk


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

FA95501110206 F1ATAO1194GOO1
