A quantile-based probabilistic mean value theorem

TL;DR

This paper introduces a novel quantile-based approach to probabilistic theorems for nonnegative random variables, extending classical results and applying them to risk theory and stochastic comparisons.

Contribution

It develops a new quantile-based probabilistic mean value theorem and a version of Taylor's theorem, along with a generalized Lorenz curve and a reversed proportional shortfall order.

Findings

New quantile-based probabilistic mean value theorem

A generalized Lorenz curve for stochastic comparisons

Introduction of the expected reversed proportional shortfall order

Abstract

For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows 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.