Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions
Antonio Di Crescenzo, Barbara Martinucci, Julio Mulero
TL;DR
The paper extends the quantile-based probabilistic mean value theorem to distorted distributions by introducing distorted random variables $X_h$ with survival $ar{F}_h(x)=h(ar{F}(x))$ and quantile $Q_h(u)=Q(1-h^{-1}(1-u))$, plus the Lorenz-curve driven variable $X^L$ with $L(p)=\frac{1}{E[X]}\int_0^p Q(u)\,du$. It derives a central identity $E\left[\left(\frac{q(1-l^{-1}(1-U))}{l'(l^{-1}(1-U))}-\frac{q(1-h^{-1}(1-U))}{h'(h^{-1}(1-U))}\right)(g(1)-g(U))\right]=E[g'(Z^L)](E[X_l]-E[X_h])$, where $f_{Z^L}(x)=\frac{Q(1-l^{-1}(1-x))-Q(1-h^{-1}(1-x))}{E[X_l]-E[X_h]}$, and explores specific distortions corresponding to the proportional hazard rate model, the proportional reversed hazard rate model, and the new-better-than-used property. By analyzing cases such as $h(t)=t^{\alpha}$ vs $l(t)=t^{\beta}$ and $h(t)=1-(1-t)^n$ vs $l(t)=1-(1-t)^m$, the paper provides explicit forms for means and the limiting distribution $Z^L$, enabling construction of new $(0,1)$-support densities and comparisons of distorted risks. These results have practical implications for utility theory and stochastic dominance assessments in actuarial contexts. Overall, the work expands the toolkit for distortion-based risk analysis and offers concrete mappings between distortions, means, and quantile-based representations.
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.