Distortion function

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A distortion function $g:[0,1]\to [0,1]$ 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

↑ 1.0 1.1 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.
↑ 2.0 2.1 Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.

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