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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:
[1] Bandemer H, Gottwald S (1993) Einführung in Fuzzy-Methoden (4th ed). Akademie Verlag Berlin · Zbl 0771.94018
[2] Bandemer H, Näther W (1992) Fuzzy data analysis. Kluwer Academic Publishers Dordrecht · Zbl 0776.94021
[3] Frühwirth-Schnatter S (1993) On fuzzy Bayesian inference. Fuzzy Sets and Systems 60:41–58 · Zbl 0796.62003
[4] Kallenberg WCM et al (1984) Testing statistical hypotheses: Worked solutions. Centrum voor Wiskunde en Informatica Amsterdam
[5] Lehmann EL (1959) Testing statistical hypotheses. Wiley New York · Zbl 0089.14102
[6] Lehmann EL (1986) Testing statistical hypotheses (2nd ed). Wiley New York · Zbl 0608.62020
[7] Rommelfanger H (1988) Entscheiden bei Unschärfe. Springer-Verlag Heidelberg · Zbl 0657.90002
[8] Saade JJ (1994) Extension of fuzzy hypothesis testing with hybrid data. Fuzzy Sets and Systems 63:57–71 · Zbl 0843.62004
[9] Uhlmann W (1982) Statistische Qualitätskontrolle (2nd ed). Teubner Verlag Stuttgart
[10] Viertl R (1991) On Bayes’ theorem for fuzzy data. Statistical Papers 32:115–122 · Zbl 0719.62011
[11] Zadeh LA (1965) Fuzzy sets. Information and Control 8:338–353 · Zbl 0139.24606
[12] Zimmermann H-J (1991) Fuzzy set theory – and its applications (2nd ed). Kluwer Academic Publishers Dordrecht · Zbl 0719.04002
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