distortion function

{{Technical|date=November 2021}}

A distortion function in mathematics and statistics, for example, $g:\; [0,1]\; \backslash to\; [0,1]$, is a non-decreasing function such that $g(0)\; =\; 0$ and $g(1)\; =\; 1$. The dual distortion function is $\backslash tilde\{g\}(x)\; =\; 1\; -\; g(1-x)$.{{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 | hdl = 10016/14071 | s2cid = 53327887 | hdl-access = free }}{{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|access-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}} Distortion functions are used to define distortion risk measures.

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

: $\backslash mathbb\{Q\}(A)\; =\; g(\backslash mathbb\{P\}(X\; \backslash in\; A)).$