# Risk function

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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:

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

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

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }} Template:Refend