Distortion function

A distortion function in <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, for example, 
 
 
 
 g
 :
 [
 0
 ,
 1
 ]
 →
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle g:[0,1]\to [0,1]}
 
, is a <a href="/facts/Non-decreasing_function/MjQK0YL2">non-decreasing function</a> 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)}
 
. Distortion functions are used to define <a href="/facts/Distortion_risk_measure/eMfxKBBb">distortion risk measures</a>.
Given a <a href="/facts/Probability_space/dEcv2VrU">probability space</a> 
 
 
 
 (
 Ω
 ,
 
 
 F
 
 
 ,
 
 P
 
 )
 
 
 {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
 
, then for any <a href="/facts/Random_variable/TwTBXnLT">random variable</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and any distortion function 
 
 
 
 g
 
 
 {\displaystyle g}
 
 we can define a new <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> 
 
 
 
 
 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)).}
 
 <a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

Distortion function open-in-new

Distortion function