View source for Risk of a statistical procedure

A characteristic giving the mean loss of an experimenter in a problem of statistical decision making and thus defining the quality of the statistical procedure under consideration.

Suppose that one has to make a decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824901.png" /> in a measurable decision space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824902.png" /> with respect to a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824903.png" /> on the basis of a realization of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824904.png" /> with values in a sampling space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824906.png" />. Further, let the loss of a statistician caused by making the decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824907.png" /> when the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824908.png" /> follows the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r0824909.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249011.png" /> is some loss function given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249012.png" />. In this case, if the statistician uses a non-randomized [[Decision-function(2)|decision function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249013.png" /> in the problem of decision making, then as a characteristic of this function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249014.png" /> the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249015.png" /></td> </tr></table>

is used. It is called the risk function or, simply, the risk, of the statistical procedure based on the decision function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249016.png" /> with respect to the loss <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249017.png" />.

The concept of risk allows one to introduce a partial order on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249018.png" /> of all non-randomized decision functions, since it is assumed that between two different decision functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249020.png" /> one should prefer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249022.png" /> uniformly over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249023.png" />.

If the decision function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249024.png" /> is randomized, the risk of the statistical procedure is defined by the formula

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082490/r08249026.png" /> is the family of Markov transition probability distributions determining the randomization procedure.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Chentsov,   "Statistical decision rules and optimal inference" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Wald,   "Statistical decision functions" , Wiley  (1950)</TD></TR></table>