An asymptotic \(\alpha\)-test for the expectation of random fuzzy variables. (English) Zbl 0976.62013

Summary: An asymptotic \(\alpha\)-test of hypotheses about the fuzzy expectation with respect to fuzzy data is obtained with the help of a central limit theorem. The asymptotical distribution of the distance between the sample mean and the fuzzy expectation is used to calculate the quantiles. The asymptotical distribution is an \(\omega^2\)-distribution. An explicit expression for the density function is obtained for tests with LR-fuzzy numbers.


62F05 Asymptotic properties of parametric tests
62E15 Exact distribution theory in statistics
62F03 Parametric hypothesis testing
03E72 Theory of fuzzy sets, etc.
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[1] Araujo, A., Giné, E., 1980. The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
[2] Dunford, N., Schwartz, J.T., 1988. Linear Operators, Part I. Wiley, New York. · Zbl 0635.47001
[3] Casals, M.R.; Gil, M.A.; Gil, P., The fuzzy decision problem: an approach of testing statistical hypotheses with fuzzy information, J. oper. res., 27, 371-382, (1986) · Zbl 0605.62018
[4] Gebhardt, J., Gil, M.A., Kruse, R., 1998. Fuzzy set-theoretical methods in statistics. In: Dubios, D., Prade, H. (Eds.), Handbook on Fuzzy Sets. Kluwer, New York. · Zbl 0946.62006
[5] Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, Proc. roy. soc. londom — ser. A, 19, 171-182, (1986) · Zbl 0605.60038
[6] Kruse, R., Meyer, K.D., 1987. Statistics with Vague Data. D. Reidel Publ. Comp., Dordrecht, Boston. · Zbl 0663.62010
[7] Martynov, G.V., 1978. Omega-Square Criteria. Nauka, Moscow (in Russian). · Zbl 0949.62531
[8] Matheron, G., 1975. Random Sets and Integral Geometry. Wiley, New York. · Zbl 0321.60009
[9] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[10] Vakhania, N.N., 1981. Probability Distribution on Linear Spaces. Elsevier Science Publishers B.V., North Holland. · Zbl 0481.60002
[11] Viertl, R., 1996. Statistical Methods for Non-Precise Data. CRC Press, Boca Raton, New York. · Zbl 0853.62004
[12] Vitale, R.A., An alternate formulation of Mean value for random geometric figures, J. microscopy, 151, 197-204, (1988)
[13] Watanabe, N.; Imaizumi, T., A fuzzy statistical test of fuzzy hypotheses, Fuzzy sets and systems, 53, 167-178, (1993) · Zbl 0795.62025
[14] Witting, H., Müller-Funk, U., 1995. Mathematische Statistik II. B.G. Teubner, Stuttgart.
[15] Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606
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