New Developments in Uncertainty: Linking Risk Management, Reliability, Statistics and Stochastic Optimization

Abstract

Random variables that stand for cost, loss or damage must be confronted in numerous situations. Dealing with them systematically for purposes in risk management, optimization and statistics is the theme of this project, which brings together ideas coming from many different areas. Measures of risk can be used to quantify the hazard in a random variable by a single value which can substitute for the otherwise uncertain outcomes in a formulation of constraints and objectives. Such quantifications of risk can be portrayed on a higher level as generated from penalty-type expressions of "regret" about the mix of potential outcomes. A trade-off between an up-front level of hazard and the uncertain residual hazard underlies that derivation. Regret is the mirror image of utility, a familiar concept for dealing with gains instead of losses, but regret concerns hazards relative to a benchmark. It bridges risk measures and expected utility, thereby reconciling those two approaches to optimization under uncertainty. Statistical estimation is inevitably a partner with risk management in handling hazards, which may be known only partially through a data base. However, a much deeper connection has come to light with statistical theory itself, in particular regression. Very general measures of error can associate with any hazard variable a "statistic" along with a "deviation" which quantifies the variable's nonconstancy. Measures of deviation, on the other hand, are paired closely with measures of risk exhibiting "aversity." A direct correspondence can furthermore be identified between measures of error and measures of regret. The fundamental quadrangle of risk developed here puts all of this together in a unified scheme.

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Document Details

Document Type
Technical Report
Publication Date
Nov 13, 2014
Accession Number
ADA616353

Entities

People

  • Stan Uryasev
  • Tyrrel R. Rockafellar

Organizations

  • University of Florida

Tags

Communities of Interest

  • C4I

DTIC Thesaurus Topics

  • Convex Sets
  • Data Science
  • Databases
  • Distribution Functions
  • Equations
  • Information Science
  • Motivation
  • Operations Research
  • Probability
  • Probability Distributions
  • Random Variables
  • Reliability Engineering
  • Risk Management
  • Statistical Analysis
  • Statistical Estimation
  • Statistics
  • Systems Engineering

Readers

  • Aviation Safety Risk Assessment.
  • Statistical inference.
  • Theoretical Analysis.