Air–fuel ratio

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:

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

where

• θ is a fixed but possibly unknown state of nature;
• X is a vector of observations stochastically drawn from a population;
• ${\mathbb {E} }_{\theta }$ is the expectation over all population values of X;
• dP_θ 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

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

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 L² norm,

$L(f,{\hat {f}})=\|f-{\hat {f}}\|_{2}^{2}\,,$

the risk function becomes the mean integrated squared error

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

References

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