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, &delta;, is the expected value of a loss function L:


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

where


 * $$\theta$$ is a fixed but possibly unknown state of nature;
 * X is a vector of observations stochastically drawn from a population;
 * $$E_{\theta}$$ is the expectation over all population values of X;
 * $$dP_\theta$$ 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 $$\theta$$, a decision function whose output $$\hat\theta$$ is an estimate of $$\theta$$, and a quadratic loss function,


 * $$L(\theta,\hat\theta)=(\theta-\hat\theta)^2$$


 * the risk function becomes the mean squared error of the estimate,


 * $$R(\theta,\hat\theta)=E_\theta(\theta-\hat\theta)^2.$$


 * In density estimation, the unknown parameter is probability density itself. The loss function is typically chosen to be a norm in an appropriate function space. For example, for $$L^2$$ norm,


 * $$L(f,\hat f)=\|f-\hat f\|_2^2\,$$


 * the risk function becomes the mean integrated squared error


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