Distortion function

A distortion function in mathematics and statistics, for example, ${\displaystyle g:[0,1]\to [0,1]}$, is a non-decreasing function such that ${\displaystyle g(0)=0}$ and ${\displaystyle g(1)=1}$. The dual distortion function is ${\displaystyle {\tilde {g}}(x)=1-g(1-x)}$.^[1][2] Distortion functions are used to define distortion risk measures.^[2]

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

${\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).}$ ^[1]