View source for Loss function

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" />.

====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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====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>