Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions

TL;DR

This paper explores applications of the quantile-based probabilistic mean value theorem to distorted distributions, focusing on actuarial models like proportional hazard and reversed hazard rate, and properties like 'new better than used'.

Contribution

It extends the theorem's application to distorted distributions under specific hazard rate models and properties, providing new insights for actuarial science.

Findings

Applications to proportional hazard rate models

Applications to proportional reversed hazard rate models

Analysis of 'new better than used' property cases

Abstract

Distorted distributions were introduced in the context of actuarial science for several variety of insurance problems. In this paper we consider the quantile-based probabilistic mean value theorem given in Di Crescenzo et al. [4] and provide some applications based on distorted random variables. Specifically, we consider the cases when the underlying random variables satisfy the proportional hazard rate model and the proportional reversed hazard rate model. A setting based on random variables having the 'new better than used' property is also analyzed.