Risk function

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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 of an estimator, $\hat\theta,$ of an unknown parameter of the distribution, $θ,$ is the expected value of the loss function

$R(\theta,\hat\theta) = {\mathbb E}_\theta L(\theta,\hat\theta)= \int L(\theta,\hat\theta) \, dP_\theta.$

where $d P θ$ is a probability measure parametrized by θ.

[edit] Examples

For a scalar parameter $θ$ 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 $L 2$ norm,

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

the risk function becomes the mean integrated squared error

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

[edit] References

James O. Berger Statistical Decision Theory and Bayesian Analysis. Second Edition. Springer-Verlag, 1980, 1985. ISBN 0-387-96098-8.
Morris De Groot Optimal Statistical Decisions. Wiley Classics Library. 2004. (Originally published 1970.) ISBN 0-471-68029-X.
Christian P. Robert The Bayesian Choice. Springer-Verlag 1994. ISBN 3-540-94296-3.