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An approach to fuzzy hypothesis testing. (English) Zbl 0862.62019
Summary: An approach is presented how to test fuzzily formulated hypotheses with crisp data. The quantities $$\alpha$$ and $$\beta$$, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions that $$\alpha$$ and $$\beta$$ can simultaneously be diminished by increasing the sample size even in the case of testing $$H_0$$ against the omnibus alternative $$H_1$$: not $$H_0$$, a result completely different from the case of crisp sets $$H_0$$ and $$H_1$$: not $$H_0$$.

##### MSC:
 62F03 Parametric hypothesis testing 62F99 Parametric inference
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##### References:
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