Distortion function explained

A distortion function in mathematics and statistics, for example,

g:[0,1]\to[0,1]

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]

(\Omega,l{F},P)

, then for any random variable

and any distortion function

we can define a new probability measure

such that for any

A\inl{F}

it follows that

Q(A)=g(P(X\inA)).

^[1]

Notes and References

Balbás . A. . Garrido . J. . Mayoral . S. . 10.1007/s11009-008-9089-z . Properties of Distortion Risk Measures . Methodology and Computing in Applied Probability . 11 . 3 . 385 . 2008 . 10016/14071 . 53327887 . free .
Web site: Distortion Risk Measures: Coherence and Stochastic Dominance. Julia L. Wirch. Mary R. Hardy. March 10, 2012. https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf. July 5, 2016. dead.