View source for Minimax estimator

A statistical estimator obtained as a result of the application of the notion of a [[Minimax statistical procedure|minimax statistical procedure]] in the problem of statistical estimation.

Example 1. Let a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639401.png" /> be subject to the binomial law with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639405.png" />, is unknown. The statistic

<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/m/m063/m063940/m0639406.png" /></td> </tr></table>

is a minimax estimator for the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639407.png" /> with respect to the loss 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/m/m063/m063940/m0639408.png" /></td> </tr></table>

Example 2. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m0639409.png" /> be independent random variables subject to the same probability law, with a continuous probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394012.png" />. The Pitman estimator

<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/m/m063/m063940/m06394013.png" /></td> </tr></table>

is a minimax estimator for the unknown shift parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394014.png" /> relative to the loss function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394016.png" /> are the order statistics (cf. [[Order statistic|Order statistic]]) obtained from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394018.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063940/m06394020.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Zacks,   "The theory of statistical inference" , Wiley  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.R. Cox,   D.V. Hinkley,   "Theoretical statistics" , Chapman &amp; Hall  (1974)</TD></TR></table>