Distortion function

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

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

\mathbb {Q} (A)=g(\mathbb {P} (X\in A)).

^[1]

References

^ ^a ^b 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.
^ ^a ^b 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.

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[PropertiesDRM-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.

[Wirch-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.

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