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Neyman-Pearson lemma for fuzzy hypotheses testing with vague data. (English) Zbl 1103.62021
Summary: In hypotheses testing, and other statistical problems, we may confront imprecise concepts. One case is a situation in which both hypotheses and observations are imprecise. This paper tries to develop a new approach for testing fuzzy hypotheses when the available data are fuzzy, too. First, some definitions are provided, such as: fuzzy sample space, fuzzy-valued random samples, and fuzzy-valued random variables. Then, the problem of fuzzy hypothesis testing with vague data is formulated. Finally, we state and prove a generalized Neyman-Pearson lemma for such problems. The proposed approach is illustrated by some numerical examples.

MSC:
62F03 Parametric hypothesis testing
03E72 Theory of fuzzy sets, etc.
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[1] Arnold BF (1996) An approach to fuzzy hypotheses testing. Metrika 44:119–126 · Zbl 0862.62019
[2] Arnold BF (1998) Testing fuzzy hypothesis with crisp data. Fuzzy Set Syst 94:323–333 · Zbl 0940.62015
[3] Arnold BF, Gerke O (2003) Testing fuzzy hypothesis in linear regression models. Metrika 57:81–95 · Zbl 1433.62206
[4] Billingsley P (1995) Probability and measure. Wiley, New York · Zbl 0822.60002
[5] Casals MR (1993) Bayesian testing of fuzzy parametric hypotheses from fuzzy information. RAIRO. Oper Res 189–199 · Zbl 0773.62001
[6] Casals MR, Gil MA, Gil P (1986) On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information. Fuzzy Set Syst 20:175–190 · Zbl 0611.62018
[7] Casals MR, Gil MA (1989) A note on the operativeness of Neyman–Pearson tests with fuzzy information. Fuzzy Set Syst 30:215–220 · Zbl 0665.62008
[8] Casella G, Berger RL (2002) Statistical inference, 2nd edn. Duxbury Press, North Scituate
[9] Delgado M, Verdegay MA, Vila MA (1985). Testing fuzzy hypotheses: a Bayesian approach. In: Gupta MM et al. (eds). Approximate reasoning in expert systems. North-Holland Publishing Co., Amsterdam, pp 307–316
[10] Filzmoser P, Viertl R (2004) Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59:21–29 · Zbl 1052.62009
[11] Grzegorzewski P (2000) Testing statistical hypotheses with vague data. Fuzzy Set Syst 112:501–510 · Zbl 0948.62010
[12] Lehmann EL (1994) Testing statistical hypotheses. Chapman-Hall, New York
[13] Liu B (2004) Uncertainty theory: an introduction to its axiomatic foundation. Physica-Verlag, Heidelberg · Zbl 1072.28012
[14] López-Díaz M, Gil MA (1997) Constructive definitions of fuzzy random variables. Stat Prob Lett 36:135–144 · Zbl 0929.60005
[15] Montenegro M, Casals MR, Lubiano MA, Gil MA (2001) Two-sample hypothesis tests of means of a fuzzy random variable. Inf Sci 133:89–100 · Zbl 1042.62012
[16] Paris MGA (2001) Nearly ideal binary communication in squeezed channels. Phys Rev A 64:14304–14308
[17] Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422 · Zbl 0592.60004
[18] Ralescu DA (1995) Fuzzy random variable revisited. In: Proceeding of the 4th IEEE International Conference on Fuzzy Systems, Yokohama, 2 993
[19] Saade J (1994) Extension of fuzzy hypotheses testing with hybrid data. Fuzzy Set Syst 63:57–71 · Zbl 0843.62004
[20] Saade J, Schwarzlander H (1990) Fuzzy hypotheses testing with hybrid data. Fuzzy Set Syst 35:192–212 · Zbl 0713.62010
[21] Son JC, Song I, Kim HY (1992) A fuzzy decision problem based on the generalized Neyman–Pearson criteria. Fuzzy Set Syst 47:65–75 · Zbl 0757.62012
[22] Taheri SM (2003) Trends in fuzzy statistics. Aust J Stat 32:239–257
[23] Taheri SM, Behboodian J (1999) Neyman–Pearson Lemma for fuzzy hypotheses testing. Metrika 49:3–17 · Zbl 1093.62520
[24] Taheri SM, Behboodian J (2001) A Bayesian approach to fuzzy hypotheses testing. Fuzzy Set Syst 123:39–48 · Zbl 0983.62015
[25] Taheri SM, Behboodian J (2002). Fuzzy hypotheses testing with fuzzy data: a Bayesian approach. In: Pal NR, Sugeno M (eds). AFSS 2002. Physica-Verlag, Heidelberg, pp 527–533 · Zbl 1108.62308
[26] Taheri SM, Behboodian J (2005) On Bayesian approach to fuzzy hypotheses testing with fuzzy data. Ital J Pure Appl Math (to appear) · Zbl 1150.62014
[27] Watanabe N, Imaizumi T (1993) A fuzzy statistical test of fuzzy hypotheses. Fuzzy Set Syst 53:167–178 · Zbl 0795.62025
[28] Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353 · Zbl 0139.24606
[29] Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427 · Zbl 0174.49002
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