# 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 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 .