A Quantile-Based Probabilistic Mean Value Theorem

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.Short title: A quantile-based probabilistic mean value theorem.

Extension of the past lifetime and its connection to the cumulative entropy

“…We conclude the paper with few illustrative examples which shed some light on the behavior of the sequences of random variables defined in (13).…”

“…We remark that a necessary and sufficient condition such that X is LBDRHR has been given in terms of stochastic comparison of quantile-based distributions in [13]. Other results on the characterization given in Definition 3 will be the object of a future investigation.…”

“…Consider the sequence of random variables {X n , n ≥ 1} as defined in (13) with the reversed hazard rate functions defined in (16), and let Y be an absolutely continuous nonnegative random variable with the reversed hazard rate function τ Y (x), x > 0. If X 1 and Y satisfy the PRHRM with (5) we have that…”

“…Remark 4 Consider the sequence of random variables as defined in (13). From (14), we have, for n ≥ 1,F…”

“…Theorem 9 For the sequence of random variables {X n , n ≥ 1} defined in (13) and for all n = 1, 2, . .…”