A Quantile-Based Probabilistic Mean Value Theorem

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.

The research of A.D.C. and B.M. has been performed under partial support by GNCS-INdAM and Regione Campania (Legge 5). J.M. was supported by project MTM2012-34023-FEDER, “Comparación y dependencia en modelos probabilísticos con aplicaciones en fiabilidad y riesgos”, from Universidad de Murcia.