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In a problem of statistical decision making, a non-negative function indicating the loss (cost) to an experimenter given a particular state of the world and a particular decision. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609001.png" /> be a random variable taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609003.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609004.png" /> be the space of all possible decisions that can be taken on the basis of an observed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609005.png" />. In the theory of statistical decision functions, any non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609006.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609007.png" /> is called a loss function. The value of a loss function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609008.png" /> at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l0609009.png" /> is interpreted as the cost incurred by taking a decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l06090010.png" />, when the true parameter is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l06090011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060900/l06090012.png" />.
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In a problem of [[Statistical decision theory|statistical decision making]], a non-negative function indicating the loss (cost) to an experimenter given a particular state of the world and a particular decision. Let $X$ be a random variable taking values in a [[sample space]] $(\mathfrak{X},\mathcal{B},\mathsf{P}_\theta)$, $\theta \in \Theta$, and let $D = \{d\}$ be the space of all possible decisions that can be taken on the basis of an observed $X$. In the theory of statistical decision functions, any non-negative function $L$ defined on $\Theta \times D$ is called a loss function. The value of a loss function $L$ at an arbitrary point $(\theta,d) \in \Theta \times D$ is interpreted as the cost incurred by taking a decision $d \in D$, when the true parameter is $\theta$, $\theta \in \Theta$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Wald,  "Statistical decision functions" , Wiley  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Wald,  "Statistical decision functions" , Wiley  (1950) {{ZBL|0040.36402}}</TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" (2nd ed.), Wiley  (1986) {{ZBL|0608.62020}}</TD></TR>
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</table>
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.O. Berger,  "Statistical decision theory and Bayesian analysis" , Springer  (1985)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.O. Berger,  "Statistical decision theory and Bayesian analysis" (2nd ed.) , Springer  (1985) {{ZBL|0572.62008}}</TD></TR>
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</table>

Latest revision as of 19:00, 12 April 2017

2020 Mathematics Subject Classification: Primary: 62C05 [MSN][ZBL]

In a problem of statistical decision making, a non-negative function indicating the loss (cost) to an experimenter given a particular state of the world and a particular decision. Let $X$ be a random variable taking values in a sample space $(\mathfrak{X},\mathcal{B},\mathsf{P}_\theta)$, $\theta \in \Theta$, and let $D = \{d\}$ be the space of all possible decisions that can be taken on the basis of an observed $X$. In the theory of statistical decision functions, any non-negative function $L$ defined on $\Theta \times D$ is called a loss function. The value of a loss function $L$ at an arbitrary point $(\theta,d) \in \Theta \times D$ is interpreted as the cost incurred by taking a decision $d \in D$, when the true parameter is $\theta$, $\theta \in \Theta$.

References

[1] A. Wald, "Statistical decision functions" , Wiley (1950) Zbl 0040.36402
[2] E.L. Lehmann, "Testing statistical hypotheses" (2nd ed.), Wiley (1986) Zbl 0608.62020

Comments

References

[a1] J.O. Berger, "Statistical decision theory and Bayesian analysis" (2nd ed.) , Springer (1985) Zbl 0572.62008
How to Cite This Entry:
Loss function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loss_function&oldid=40954
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article