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One of the versions of optimality in mathematical statistics, according to which a statistical procedure is pronounced optimal in the minimax sense if it minimizes the maximal risk. In terms of decision functions (cf. [[Decision-function(2)|Decision function]]) the notion of a minimax statistical procedure is defined as follows. Let a random variable  $  X $
take values in a sampling space  $  ( \mathfrak X , \mathfrak B , {\mathsf P} _  \theta  ) $, 
$  \theta \in \Theta $, 
and let  $  \Delta = \{ \delta \} $
be the class of decision functions which are used to make a decision  $  d $
from the decision space  $  D $
on the basis of a realization of  $  X $, 
that is,  $  \delta ( \cdot ) :  \mathfrak X \rightarrow D $. 
In this connection, the loss function  $  L ( \theta , d) $, 
defined on  $  \Theta \times D $, 
is assumed given. In such a case a statistical procedure  $  \delta  ^ {*} \in \Delta $
is called a minimax procedure in the problem of making a statistical decision relative to the loss function  $  L ( \theta , d ) $
if for all  $  \delta \in \Delta $,

$$ 
\sup _ {\theta \in \Theta } \ 
{\mathsf E} _  \theta  L ( \theta , \delta  ^ {*} ( X) )
 \leq   \sup _ {\theta \in \Theta } \ 
{\mathsf E} _  \theta  L ( \theta , \delta ( X) ) ,
$$

where

$$ 
{\mathsf E} _  \theta  L ( \theta , \delta ( X) )  = \ 
R ( \theta , \delta )  = \ 
\int\limits _ { \mathfrak X } L ( \theta , \delta ( X) ) \ 
d {\mathsf P} _  \theta  ( x)
$$

is the risk function associated to the statistical procedure (decision rule)  $  \delta $; 
the decision  $  d  ^ {*} = \delta  ^ {*} ( x) $
corresponding to an observation  $  x $
and the minimax procedure  $  \delta  ^ {*} $
is called the minimax decision. Since the quantity

$$ 
\sup _ {\theta \in \Theta } \ 
{\mathsf E} _  \theta  L ( \theta , \delta ( X) )
$$

shows the expected loss under the procedure  $  \delta \in \Delta $, 
$  \delta  ^ {*} $
being maximal means that if  $  \delta  ^ {*} $
is used to choose a decision  $  d $
from  $  D $, 
then the largest expected risk,

$$ 
\sup _ {\theta \in \Theta } \ 
R ( \theta , \delta  ^ {*} ) ,
$$

will be as small as possible.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m063970a.gif" />

Figure: m063970a

The minimax principle for a statistical procedure does not always lead to a reasonable conclusion (see Fig. a); in this case one must be guided by  $  \delta _ {1} $
and not by  $  \delta _ {2} $, 
although

$$ 
\sup _ {\theta \in \Theta } \ 
R ( \theta , \delta _ {1} )  > \ 
\sup _ {\theta \in \Theta } \ 
R ( \theta , \delta _ {2} ) .
$$

The notion of a minimax statistical procedure is useful in problems of statistical decision making in the absence of a priori information regarding  $  \theta $.

====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">  S. Zacks,   "The theory of statistical inference" , Wiley  (1971)</TD></TR></table>