A distortion function in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, for example,

g:,1 \to,1 /math>, is a

non-decreasing function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

such that

g(0) = 0

and

g(1) = 1

. The dual distortion function is

\tilde(x) = 1 - g(1-x)

. Distortion functions are used to define

distortion risk measure In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio. Mathematical definition The function \rho_g: L^p \to \m ...

s. Given a

probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

(\Omega,\mathcal,\mathbb)

, then for any

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

X

and any distortion function

g

we can define a new

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

\mathbb

such that for any

A \in \mathcal

it follows that :

\mathbb(A) = g(\mathbb(X \in A)).

References

Functions related to probability distributions {{probability-stub