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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]].''
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| In [[decision theory]] and [[estimation theory]], the '''risk function''' ''R'' of a [[decision rule]] ''δ'', is the [[expected value]] of a [[loss function]] ''L'':
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| :<math> R(\theta,\delta) = {\mathbb E}_\theta L\big(\theta,\delta(X) \big)= \int_\mathcal{X} L\big( \theta,\delta(X) \big) \, dP_\theta(X)</math>
| | I'm a 31 years old and working at the college ([http://browse.deviantart.com/?qh=§ion=&global=1&q=Integrated+International Integrated International] Studies).<br>In my spare time I learn Arabic. I have been there and look forward to go there anytime soon. I like to read, preferably on my kindle. I really love to watch Doctor Who and Supernatural as well as documentaries about anything [http://www.Squidoo.com/search/results?q=astronomical astronomical]. I like Radio-Controlled Car Racing.<br><br>Here is my web site Fifa 15 Coin Generator ([http://nickodelguercio.99k.org/2/forum/index.php?a=member&m=13937 nickodelguercio.99k.org]) |
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| where
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| *''θ'' is a fixed but possibly unknown state of nature;
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| *''X'' is a vector of observations stochastically drawn from a population;
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| *<math>{\mathbb E}_\theta</math> is the expectation over all population values of ''X'';
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| *''dP<sub>θ</sub>'' is a [[probability measure]] over the event space of ''X'', parametrized by ''θ''; and
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| *the integral is evaluated over the entire support of ''X''.
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| ==Examples== | |
| * For a scalar parameter ''θ'', a decision function whose output <math>\hat\theta</math> is an estimate of ''θ'', and a quadratic loss function
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| ::<math>L(\theta,\hat\theta)=(\theta-\hat\theta)^2,</math>
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| :the risk function becomes the [[mean squared error]] of the estimate,
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| ::<math>R(\theta,\hat\theta)=E_\theta(\theta-\hat\theta)^2.</math>
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| * In [[density estimation]], the unknown parameter is [[probability density function|probability density]] itself. The loss function is typically chosen to be a norm in an appropriate [[function space]]. For example, for ''L<sup>2</sup>'' norm,
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| ::<math>L(f,\hat f)=\|f-\hat f\|_2^2\,,</math>
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| :the risk function becomes the [[mean integrated squared error]]
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| ::<math>R(f,\hat f)=E \|f-\hat f\|^2.\,</math> | |
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| ==References==
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| {{refbegin}}
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| * {{SpringerEOM| title=Risk of a statistical procedure |id=R/r082490 |first=M. S. |last=Nikulin}}
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| * {{cite book
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| |title=Statistical decision theory and Bayesian Analysis
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| |first=James O. |last=Berger |authorlink=James Berger (statistician)
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| |year=1985
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| |edition=2nd
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| |publisher=Springer-Verlag |location=New York
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| |ISBN=0-387-96098-8 |mr=0804611
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| }}
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| * {{cite book
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| |first=Morris |last=DeGroot |authorlink=Morris H. DeGroot
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| |title=Optimal Statistical Decisions
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| |publisher=Wiley Classics Library |year=2004 |origyear=1970
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| |ISBN=0-471-68029-X |mr=2288194
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| }}
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| * {{cite book
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| |last=Robert |first=Christian P.
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| |title=The Bayesian Choice
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| |publisher=Springer |location=New York
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| |year=2007|edition=2nd
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| |doi=10.1007/0-387-71599-1 |isbn=0-387-95231-4 |mr=1835885
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| }}
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| {{refend}}
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| [[Category:Decision theory]]
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| [[Category:Estimation theory]]
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| {{statistics-stub}}
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I'm a 31 years old and working at the college (Integrated International Studies).
In my spare time I learn Arabic. I have been there and look forward to go there anytime soon. I like to read, preferably on my kindle. I really love to watch Doctor Who and Supernatural as well as documentaries about anything astronomical. I like Radio-Controlled Car Racing.
Here is my web site Fifa 15 Coin Generator (nickodelguercio.99k.org)