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Realization of closed, convex, and symmetric subsets of the unit square as regions of risk for testing simple hypotheses. (English) Zbl 0834.62006
Summary: It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and \((1/2,1/2)\)-symmetric subset of the unit square, which contains the points \((0,0)\) and \((1,1)\). It is shown that for any such subset \(R\) of the unit square and any atomless probability measure \(P\) on some \(\sigma\)-algebra there exists some probability measure \(Q\) on the same \(\sigma\)-algebra such that \(R\) is the corresponding region of risk for testing \(P\) against \(Q\).
This generalizes a result of W. Sendler [Z. Wahrscheinlichkeitstheorie Verw. Geb. 18, 183-196 (1971; Zbl 0196.217)] and is as a first step derived here for the special case, where \(P\) is equal to the uniform distribution on the unit interval. The corresponding distribution \(Q\) is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows that \(Q\) might be chosen to be of type \(Q=\lambda Q_1+(1- \lambda)Q_2\) for some \(\lambda\) satisfying \(0\leq\lambda\leq 1\), where \(Q_1\) is a probability measure, which is absolutely continuous with respect to \(P\) and \(Q_2\) is a one-point mass.
MSC:
62B99 Sufficiency and information
62F03 Parametric hypothesis testing
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References:
[1] Halmos P (1974) Measure theory. Springer New York · Zbl 0283.28001
[2] Nemetz T (1977) Information type measures and their application fo finite decision problems. Carleton Mathematical Lecture Note 17 · Zbl 0354.62010
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[4] Sendler W (1971) Einige maßtheoretische Sätze bei der Behandlung trennscharfer Tests. Z Wahrscheinlichkeitstheorie verw Geb 18:183–196 · Zbl 0207.18501
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