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A `!distortion function`! in `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f and `F33f`_`[statistics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Statistics]`_`f, for example, g : [ 0 , 1 ] → → [ 0 , 1 ] {\\displaystyle g:[0,1]\\to [0,1]} , is a `F33f`_`[non-decreasing function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-decreasing_function]`_`f 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[`F5bf`_`[1`#cite-note-propertiesdrm-1]`_`f]`:cite-ref-wirch-2-0[`F5bf`_`[2`#cite-note-wirch-2]`_`f] Distortion functions are used to define `F33f`_`[distortion risk measures`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Distortion_risk_measure]`_`f.`:cite-ref-wirch-2-1[`F5bf`_`[2`#cite-note-wirch-2]`_`f]

Given a `F33f`_`[probability space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_space]`_`f ( Ω Ω , F , P ) {\\displaystyle (\\Omega ,{\\mathcal {F}},\\mathbb {P} )} , then for any `F33f`_`[random variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_variable]`_`f X {\\displaystyle X} and any distortion function g {\\displaystyle g} we can define a new `F33f`_`[probability measure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_measure]`_`f 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[`F5bf`_`[1`#cite-note-propertiesdrm-1]`_`f]

>>References

`:cite-note-propertiesdrm-1`!1.`! `F0af`_`[↑`#cite-ref-propertiesdrm-1-0]`_`f `:citerefbalb-sgarridomayoral2008`aBalbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". `*Methodology and Computing in Applied Probability`*. `!11`! (3): 385. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/s11009-008-9089-z. `F33f`_`[hdl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hdl_(identifier)]`_`f:10016/14071. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 53327887.

`:cite-note-wirch-2`!2.`! `F0af`_`[↑`#cite-ref-wirch-2-0]`_`f `:citerefjulia-l-wirchmary-r-hardy`aJulia 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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