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A QUANTILE-BASED PROBABILISTIC MEAN VALUE THEOREM

Published online by Cambridge University Press:  09 December 2015

Antonio Di Crescenzo ,
Barbara Martinucci  and
Julio Mulero
Antonio Di Crescenzo
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132; 84084 Fisciano (SA), Italy E-mail: adicrescenzo@unisa.it
Barbara Martinucci
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132; 84084 Fisciano (SA), Italy E-mail: bmartinucci@unisa.it
Julio Mulero
Affiliation:
Departamento de Matemáticas, Universidad de Alicante, Apartado de Correos, 99; 03080 Alicante, Spain E-mail: julio.mulero@ua.es
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Abstract

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 261 - 280
Copyright
Copyright © Cambridge University Press 2015 

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