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, 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;
• E_θ 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

• Nikulin, M.S. (2001), "Risk of a statistical procedure", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=R/r082490 
• Berger, James O. (1985). Statistical decision theory and Bayesian Analysis (2nd ed.). New York: Springer-Verlag. ISBN 0-387-96098-8. MR0804611. 
• DeGroot, Morris (2004) [1970]. Optimal Statistical Decisions. Wiley Classics Library. ISBN 0-471-68029-X. MR2288194. 
• Robert, Christian (2007). The Bayesian Choice (2nd ed.). New York: Springer. doi:10.1007/0-387-71599-1. ISBN 0-387-95231-4. MR1835885.