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 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

$\theta$ 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.\,

References[]

Template:SpringerEOM
Berger, James O. (1985). Statistical decision theory and Bayesian Analysis, 2nd, New York: Springer-Verlag.
DeGroot, Morris [1970] (2004). Optimal Statistical Decisions, Wiley Classics Library.
Robert, Christian (2007). The Bayesian Choice, 2nd, New York: Springer.

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