Editing Loss function (section)

====Frequentist expected loss====
We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the [[probability distribution]], ''P''<sub>''θ''</sub>, of the observed data, ''X''. This is also referred to as the '''risk function'''<ref>{{SpringerEOM| title=Risk of a statistical procedure |id=R/r082490 |first=M.S. |last=Nikulin}}</ref><ref>
{{cite book
 |title=Statistical decision theory and Bayesian Analysis
 |first=James O. |last=Berger |author-link=James Berger (statistician)
 |year=1985
 |edition=2nd
 |publisher=Springer-Verlag |location=New York
 |isbn=978-0-387-96098-2 |mr=0804611
|url=https://books.google.com/books?id=oY_x7dE15_AC
|bibcode=1985sdtb.book.....B }}</ref><ref>{{cite book
 |first=Morris |last=DeGroot |author-link=Morris H. DeGroot
 |title=Optimal Statistical Decisions
 |publisher=Wiley Classics Library |year=2004 |orig-year=1970
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}}</ref><ref>{{cite book
 |last=Robert |first=Christian P.
 |title=The Bayesian Choice
 |publisher=Springer |location=New York
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 |doi=10.1007/0-387-71599-1 |isbn=978-0-387-95231-4 |mr=1835885
|series=Springer Texts in Statistics
 }}</ref> of the decision rule ''δ'' and the parameter ''θ''. Here the decision rule depends on the outcome of ''X''. The risk function is given by:

: <math>R(\theta, \delta) = \operatorname{E}_\theta L\big( \theta, \delta(X) \big) = \int_X L\big( \theta, \delta(x) \big) \, \mathrm{d} P_\theta (x) .</math>

Here, ''θ'' is a fixed but possibly unknown state of nature, ''X'' is a vector of observations stochastically drawn from a [[Statistical population|population]], <math>\operatorname{E}_\theta</math> is the expectation over all population values of ''X'', ''dP''<sub>''θ''</sub> is a [[probability measure]] over the event space of ''X'' (parametrized by&nbsp;''θ'') and the integral is evaluated over the entire [[Support (measure theory)|support]] of&nbsp;''X''.