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# TV2006 - c 2006 Society for Industrial and Applied...

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T HEORY P ROBAB. A PPL . c 2006 Society for Industrial and Applied Mathematics Vol. 50, No. 3, pp. 463–466 Translated from Russian Journal ON THE DVORETZKY–WALD–WOLFOWITZ THEOREM ON NONRANDOMIZED STATISTICAL DECISIONS E. A. FEINBERG AND A. B. PIUNOVSKIY ( Translated by the authors ) Abstract. The result by Dvoretzky, Wald, and Wolfowitz on the suﬃciency of nonrandomized decision rules for statistical decision problems with nonatomic state distributions holds for arbitrary Borel decision sets and for arbitrary measurable loss functions. Key words. decision rule, nonatomic measure, nonrandomized decision rule, equivalent decision rules DOI. 10.1137/S0040585X97981937 Consider the following statistical decision problem studied by Dvoretzky, Wald, and Wolfowitz [5], [6]. Let ( X, X ) be a Borel space and let Ω = { μ 1 , μ 2 , . . . , μ N } be a finite number of probability measures on ( X, X ). The true probability distribution μ on ( X, X ) is not known, but it is known that μ Ω. There is also given a Borel space ( A, A ) whose elements a represent the possible decisions. Let A ( x ) ∈ A be the sets of decisions allowable at states x X . We assume that (a) the graph Gr ( A ) = { ( x, a ): x X, a A ( x ) } is a measurable subset of X × A , and (b) there exists at least one measurable mapping ϕ : X A with ϕ ( x ) A ( x ) for all x X . In particular, (b) implies that A ( x ) = , x X . A decision rule π is a regular transition probability from X to A such that π ( A ( x ) | x ) = 1 for all x X . A decision rule is called nonrandomized if for each x X the measure π ( ·| x ) is concentrated at one point. A nonrandomized decision rule π is defined by a measurable mapping ϕ : X A such that ϕ ( x ) A ( x ) and π ( ϕ ( x ) | x ) = 1, x X . We call such a mapping a decision function and denote it by ϕ . The loss is defined by a vector-function ρ ( μ, x, a ) = ( ρ 1 ( μ, x, a ) , . . . , ρ M ( μ, x, a )) , where μ is the true value from Ω and a A ( x ). For each μ n Ω, the function ρ m ( μ n , x, a ), m = 1 , . . . , M , is measurable in ( x, a ) and takes values in [ −∞ , ]. For any decision rule π and any μ n Ω, consider the risk vector R ( μ n , π ) = ( R 1 ( μ n , π ) , . . . , R M ( μ n , π )) , where R m ( μ n , π ) = X A ρ m ( μ n , x, a ) π ( da | x ) μ n ( dx ) , m = 1 , . . . , M. Here and in what follows, all the integrals of a measurable function f are defined as f ( y ) ν ( dy ) = F + + F , where + + ( −∞ ) = −∞ , F ± = f ± ( y ) ν ( dy ), f + = max { f, 0 } , and f = min { f, 0 } .

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