On some measures and distances for positive random variables

• Published in: Applied stochastic models in business and industry 2003-04, Vol.19 (2), p.133-146
• Main Author: Finkelstein, M. S.
• Format: Article
• Language: English
• Subjects: accelerated life model; Applications; Exact sciences and technology; expected shortfall; Inference from stochastic processes; time series analysis; Mathematics; measures of risk; Probability and statistics; Probability theory and stochastic processes; proportional hazards model; random environment; Reliability, life testing, quality control; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistics; stochastic ordering; Stochastic processes; stop-loss
• ISSN: 1524-1904; 1526-4025
• DOI: 10.1002/asmb.491

A number of conventional measures of risk as real‐valued functions on the space of positive random variables are considered: the expected shortfall, the mean excess over the threshold, the stop‐loss and some others. Ordering of risks, based on these measures and the distances between corresponding distribution functions, are described. The perturbed measures, describing the effect of changing environment, are discussed. These measures are defined by the accelerated life and proportional hazards models widely used in reliability and survival analysis. The case of a random environment is of a prime interest in the paper. The main result states that if, for instance, the stochastic environment is ‘neutral in expectation’ with respect to the baseline one, the distance between the corresponding distribution functions can be still sufficiently large. Copyright © 2003 John Wiley & Sons, Ltd.