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]

References

1. Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability. 11 (3): 385. doi:10.1007/s11009-008-9089-z. hdl:10016/14071. S2CID 53327887.
2. Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (PDF). Archived from the original (PDF) on July 5, 2016. Retrieved March 10, 2012.