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Minimax statistical procedure

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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) the notion of a minimax statistical procedure is defined as follows. Let a random variable take values in a sampling space , , and let be the class of decision functions which are used to make a decision from the decision space on the basis of a realization of , that is, . In this connection, the loss function , defined on , is assumed given. In such a case a statistical procedure is called a minimax procedure in the problem of making a statistical decision relative to the loss function if for all ,

where

is the risk function associated to the statistical procedure (decision rule) ; the decision corresponding to an observation and the minimax procedure is called the minimax decision. Since the quantity

shows the expected loss under the procedure , being maximal means that if is used to choose a decision from , then the largest expected risk,

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 and not by , although

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

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. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Minimax_statistical_procedure&oldid=15340
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098