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A Theorem About the Existence of Minimax Rules for Statistical Decision Problems with Trapezoidal Fuzzy Losses
This paper considers the problem of choice of an optimal decision under unknown state of nature when losses are not known exactly for some or all decisions and some or all states of nature. However, it is possible to determine probabilities for all states of nature on the basis of statistical data. So, there are two kinds of uncertainty, randomness and vagueness. That is why both methods of the probability theory and methods of the theory of fuzzy sets are used. There is no a unified approach to choice of an optimal decision even in the classical statistical decision theory. One of existing approaches is to use minimax decision rules when losses are crisp. In this paper, the approach is generalized for problems with trapezoidal fuzzy losses. A theorem about the existence of a minimax decision rule is proved when the set of all possible states of nature is finite.
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