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| 概要 |
In the present paper, we consider a statistical decision problem as follows. We observe $ n $ samples $ X_1, cdots, X_n $ from a population and choose an action $ a $ according to the samples, where t...he action $ a $ is prescribing a decision for some population characteristic $ \theta $, and the loss for $ a $ is given by $ L(\theta, a) $. And we assume that the distribution function of $ X=(X_1, cdots, X_n)^t $ is not only depending on $ \theta $ but also on some other population characteristic $ \xi $. Since our decision problem is just for parameter $ \theta $ but not for $ \xi $, so $ \xi $ is called a nuisance parameter usually. In such a decision problem, the average loss function and Bayes risk function depend on the value of $ \xi $. Therefore, in the paper, what we want to study in this case is to find out relations between the risk functions for the decision problem with a fixed nuisance parameter $ \xi $ and one with a prior distribution of $ \xi $, and also find out relations between optimal (in some sense) decision functions for the above two different statistical decision problems. For some special case, the above problems were approached in the way of finding sufficient conditions for that the optimal decision function for fixed value $ \xi $ be independent of the value of $ \xi $. In §2 we introduce the necessary notations and preliminaries, and also have lemmas which give the relations of the risk functions. In §3 we define a sufficient statistic for a class of distributions with nuisance parameters according to D. A. S. Fraser [2] and show the completeness of the class of all decision functions based on the sufficient statistic. In §4 we consider an invariant decision problem under some transformation group on the sample space. Finally, in §5, we give the relations between optimal (minimax or Bayes sense) decision functions for the above two different decision problems. For the sake of avoiding the mathematical complexity we assume appropriate measurability and integrability of functions in the present paper.続きを見る
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