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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
##### Keywords:
simple hypotheses; atomless probability measures
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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 [3] Roberts A, Varberg D (1973) Convex functions. Academic Press New York · Zbl 0271.26009 [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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