# Risk function

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

${\displaystyle R(\theta ,\delta )={\mathbb {E} }_{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{\mathcal {X}}L{\big (}\theta ,\delta (X){\big )}\,dP_{\theta }(X)}$

where

## Examples

${\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},}$
the risk function becomes the mean squared error of the estimate,
${\displaystyle R(\theta ,{\hat {\theta }})=E_{\theta }(\theta -{\hat {\theta }})^{2}.}$
${\displaystyle L(f,{\hat {f}})=\|f-{\hat {f}}\|_{2}^{2}\,,}$
the risk function becomes the mean integrated squared error
${\displaystyle R(f,{\hat {f}})=E\|f-{\hat {f}}\|^{2}.\,}$

## References

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