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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:


  • is a fixed but possibly unknown state of nature;
  • X is a vector of observations stochastically drawn from a population;
  • is the expectation over all population values of X;
  • 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[edit | edit source]

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

References[edit | edit source]

  • 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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