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# Risk function

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### Risk function

This article is about the mathematical definition of risk in statistical decision theory. For a more general discussion of concepts and definitions of risk, see the main article Risk.

In decision theory and estimation theory, the risk function R of a decision rule δ, is the expected value of a loss function L:

$R\left(\theta,\delta\right) = \left\{\mathbb E\right\}_\theta L\big\left(\theta,\delta\left(X\right) \big\right)= \int_\mathcal\left\{X\right\} L\big\left( \theta,\delta\left(X\right) \big\right) \, dP_\theta\left(X\right)$

where

• θ is a fixed but possibly unknown state of nature;
• X is a vector of observations stochastically drawn from a population;
• $\left\{\mathbb E\right\}_\theta$ is the expectation over all population values of X;
• dPθ is a probability measure over the event space of X, parametrized by θ; and
• the integral is evaluated over the entire support of X.

## Examples

• For a scalar parameter θ, a decision function whose output $\hat\theta$ is an estimate of θ, and a quadratic loss function
$L\left(\theta,\hat\theta\right)=\left(\theta-\hat\theta\right)^2,$
the risk function becomes the mean squared error of the estimate,
$R\left(\theta,\hat\theta\right)=E_\theta\left(\theta-\hat\theta\right)^2.$
$L\left(f,\hat f\right)=\|f-\hat f\|_2^2\,,$
the risk function becomes the mean integrated squared error
$R\left(f,\hat f\right)=E \|f-\hat f\|^2.\,$

## References

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