# Risk function

Revision as of 14:35, 1 April 2013 by en>Cherkash (simplified inline math)

*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*; *dP*is a probability measure over the event space of_{θ}*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 is an estimate of*θ*, and a quadratic loss function

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

- 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*norm,^{2}

- the risk function becomes the mean integrated squared error

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }} Template:Refend