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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)


  • \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.


  • For a scalar parameter \theta, a decision function whose output \hat\theta is an estimate of \theta, and a quadratic loss function,
the risk function becomes the mean squared error of the estimate,
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.\,


  • 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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This page uses Creative Commons Licensed content from Wikipedia (view authors).

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