Distortion function: Difference between revisions

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{{contextTechnical|date=MayNovember 20122021}}

A '''distortion function''' in [[mathematics]] and [[statistics]], for example, <math>g: [0,1] \to [0,1]</math>, is a [[non-decreasing function]] such that <math>g(0) = 0</math> and <math>g(1) = 1</math>. The '''dual distortion function''' is <math>\tilde{g}(x) = 1 - g(1-x)</math>.<ref name="PropertiesDRM">{{Cite journal | last1 = Balbás | first1 = A. | last2 = Garrido | first2 = J. | last3 = Mayoral | first3 = S. | doi = 10.1007/s11009-008-9089-z | title = Properties of Distortion Risk Measures | journal = Methodology and Computing in Applied Probability | volume = 11 | issue = 3 | pages = 385 | year = 2008 | pmidhdl = 10016/14071 | pmcs2cid = 53327887 | hdl-access = free }}</ref><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|accessdateaccess-date=March 10, 2012|archive-url=https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|archive-date=July 5, 2016|url-status=dead}}</ref> Distortion functions are used to define [[distortion risk measure]]s.<ref name="Wirch"/>

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