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:

where

  • θ 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;
  • dPθ 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

the risk function becomes the mean squared error of the estimate,
the risk function becomes the mean integrated squared error

References

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