Minimax statistical procedure

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) 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.

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

 [1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) [2] S. Zacks, "The theory of statistical inference" , Wiley (1971)
How to Cite This Entry:
Minimax statistical procedure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimax_statistical_procedure&oldid=47846
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article