Distortion function

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 2 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.
1 2 Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.

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