zbMATH — the first resource for mathematics

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.

62F03 Parametric hypothesis testing
03E72 Theory of fuzzy sets, etc.
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.