Distortion function

Template:Technical A distortion function in mathematics and statistics, for example, $g:[0,1]\to [0,1]$, is a non-decreasing function such that $g(0)=0$ and $g(1)=1$. The dual distortion function is $\tilde{g}(x)=1-g(1-x)$. Distortion functions are used to define distortion risk measures.

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

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

References

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