Distortion function

A distortion function in mathematics and statistics, for example, [math]\displaystyle{ g: [0,1] \to [0,1] }[/math], is a non-decreasing function such that [math]\displaystyle{ g(0) = 0 }[/math] and [math]\displaystyle{ g(1) = 1 }[/math]. The dual distortion function is [math]\displaystyle{ \tilde{g}(x) = 1 - g(1-x) }[/math]. Distortion functions are used to define distortion risk measures.

Given a probability space [math]\displaystyle{ (\Omega,\mathcal{F},\mathbb{P}) }[/math], then for any random variable [math]\displaystyle{ X }[/math] and any distortion function [math]\displaystyle{ g }[/math] we can define a new probability measure [math]\displaystyle{ \mathbb{Q} }[/math] such that for any [math]\displaystyle{ A \in \mathcal{F} }[/math] it follows that

[math]\displaystyle{ \mathbb{Q}(A) = g(\mathbb{P}(X \in A)). }[/math]