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]


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