On the Pitman Closeness Criterion from the Decision-Theoretic Point of View
نویسنده
چکیده
The Pitman closeness ordering of statistical decision rules is considered in the general framework of comparative risk functions. The notions of admis-sibility and of the Bayes rule in this setting (C{admissibility and C{Bayes rule) are introduced. C{admissibility of C{Bayes procedures is demonstrated along with the geometric interpretation of these properties. It is shown that for general comparative risk functions the minimal complete class can contain C{inadmissible rules. Conditions for extended C{admissibility of the best equivariant rule in an invariant statistical problem are also given. 1 Comparative Risk Functions: Introduction Consider a statistical decision problem involving parameter and possible actions d from the action space A. Let P be a member of the underlying family of probability distributions of observation X, and let w(; d) be a loss function. According to the classical concept of Pitman, a statistical decision function 0 (X) is Pitman closer to than 1 (X) if
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