 #jsDisabledContent { display:none; } My Account | Register | Help Flag as Inappropriate This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate?          Excessive Violence          Sexual Content          Political / Social Email this Article Email Address:

# Risk function

Article Id: WHEBN0001848177
Reproduction Date:

 Title: Risk function Author: World Heritage Encyclopedia Language: English Subject: List of statistics articles Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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:

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

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
$R\left(f,\hat f\right)=E \|f-\hat f\|^2.\,$