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

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

Q ( A ) = g ( P ( X ∈ ∈ A ) ) . {\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).} cite-ref-propertiesdrm-1-1[1]

cite-note-propertiesdrm-11. ↑ citerefbalb-sgarridomayoral2008Balbás, 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.

cite-note-wirch-22. ↑ citerefjulia-l-wirchmary-r-hardyJulia 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.