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Overview on the development of fuzzy random variables. (English) Zbl 1108.60006
The paper presents a brief overview on probability theory and statistics for fuzzy random variables (FRV). Two main approaches are discussed: the Kwakernaak approach where an FRV is considered as a fuzzy perception of a real random variable, and the Puri-Ralescu approach where an FRV is considered as a variable with fuzzy set-valued outcomes. From the large number of special results on FRV’s the authors discuss in some detail a law of large numbers, a central limit theorem and a special result on testing hypotheses for the Aumann expectation.

MSC:
60B99Probability theory on general structures
03E72Fuzzy set theory
26E50Fuzzy real analysis
60D05Geometric probability and stochastic geometry
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References:
[1] Bertoluzza, C.; Corral, N.; Salas, A.: On a new class of distances between fuzzy numbers. Mathware & soft comput. 2, 71-84 (1995) · Zbl 0887.04003
[2] A. Colubi, Strong laws of large numbers for fuzzy random variables, Ph.D. Thesis, Universidad de Oviedo, 2000.
[3] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Gil, M. A.: A generalized strong law of large numbers. Probab. theory rel. Fields 114, 401-417 (1999) · Zbl 0933.60023
[4] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Ralescu, D. A.: On the formalization of fuzzy random variables. Inform. sci. 133, 3-6 (2001) · Zbl 0988.28008
[5] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Ralescu, D. A.: A DE[0,1]-representation of random upper semicontinuous functions. Proc. amer. Math. soc. 130, 3237-3242 (2002) · Zbl 1005.28003
[6] Colubi, A.; Fernández-García, C.; Gil, M. A.: Simulation of random fuzzy variables: an empirical approach to statistical/probabilistic studies with fuzzy experimental data. IEEE trans. Fuzzy systems 10, 384-390 (2002)
[7] Diamond, P.: Least squares Fitting of compact set-valued data. J. math. Anal. appl. 147, 351-362 (1990) · Zbl 0704.65006
[8] Feng, Y.: Decomposition theorems for fuzzy supermartingales and submartingales. Fuzzy sets and systems 116, 225-235 (2000) · Zbl 0966.60042
[9] Gil, M. A.; López-Díaz, M.: Fundamentals and Bayesian analyses of decision problems with fuzzy-valued utilities. Intenat. J. Approx. reasoning 15, 203-224 (1996) · Zbl 0949.91504
[10] Gil, M. A.; López-Díaz, M.; López-García, H.: The fuzzy hyperbolic inequality index associated with fuzzy random variables. European J. Oper. res. 110, 377-391 (1998) · Zbl 0938.91044
[11] M.A. Gil, M. Montenegro, G. González-Rodríguez, A. Colubi, M.R. Casals, Bootstrap approach to the multi-sample test of means with imprecise data, Comp. Statist. Data Anal. 2006, in press. · Zbl 1157.62391
[12] Gong, Z.; Wu, C.; Li, B.: On the problem of characterizing derivatives for the fuzzy-valued functions. Fuzzy sets and systems 127, 315-322 (2002) · Zbl 0995.26018
[13] G. González-Rodríguez, M. Montenegro, A. Colubi, M.A. Gil, Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data, Fuzzy Sets and Systems, (2006) (in this issue). · Zbl 1119.62037
[14] Hirota, K.: Probabilistic sets: probabilistic extensions of fuzzy sets. An introduction to fuzzy logic applications in intelligent systems, kluwer international series in engineering and computer science 165, 335-354 (1992)
[15] Inoue, H.; Taylor, R. L.: A SLLN for arrays of rowwise exchangeable fuzzy random sets. Stochastic anal. Appl. 13, 461-470 (1995) · Zbl 0832.60043
[16] Kim, Y. -K.: Measurability for fuzzy valued functions. Fuzzy sets and systems 129, 105-109 (2002) · Zbl 1003.28013
[17] Klement, E. P.; Puri, M. L.; Ralescu, D. A.: Law of large numbers and central limit theorems for fuzzy random variables. Cybernetics and systems research 2, 525-529 (1984) · Zbl 0549.60033
[18] Klement, E. P.; Puri, M. L.; Ralescu, D. A.: Limit theorems for fuzzy random variables. Proc. roy. Soc. London A 407, 171-182 (1986) · Zbl 0605.60038
[19] Körner, R.: On the variance of fuzzy random variables. Fuzzy sets and systems 92, 83-93 (1997) · Zbl 0936.60017
[20] Körner, R.: An asymptotic $\alpha $-test for the expectation of random fuzzy variables. J. statist. Plann. inference 83, 331-346 (2000) · Zbl 0976.62013
[21] Körner, R.; Näther, W.: Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates. Inform. sci. 109, 95-118 (1998) · Zbl 0930.62072
[22] Körner, R.; Näther, W.: On the variance of random fuzzy variables. Statistical modeling, 22-39 (2002)
[23] Krätschmer, V.: Limit theorems for fuzzy-random variables. Fuzzy sets and systems 126, 253-263 (2002) · Zbl 0996.60042
[24] V. Krätschmer, Integrals of random fuzzy sets, Abst. 25th Linz Seminar on Fuzzy Set Theory 2004, pp. 93 -- 103.
[25] R. Kruse, K.D. Meyer, Statistics with Vague Data, Reidel Publ. Co. Dordrecht, 1987. · Zbl 0663.62010
[26] Kwakernaak, H.: Fuzzy random variables. Part I: Definitions and theorems. Inform. sci. 15, 1-29 (1978) · Zbl 0438.60004
[27] Kwakernaak, H.: Fuzzy random variables. Part II: Algorithms and examples for the discrete case. Inform. sci. 17, 253-278 (1979) · Zbl 0438.60005
[28] Li, S.; Ogura, Y.: A convergence theorem of fuzzy-valued martingales in the extended Hausdorff metric H$\infty $. Fuzzy sets and systems 135, 391-399 (2003) · Zbl 1020.60036
[29] Li, S.; Ogura, Y.; Kreinovich, V.: Limit theorems and applications of set valued and fuzzy valued random variables. (2002)
[30] Li, S.; Ogura, Y.; Nguyen, H. T.: Gaussian processes and martingales for fuzzy valued random variables with continuous parameter. Inform. sci. 133, 7-21 (2001) · Zbl 0988.60025
[31] Li, S.; Ogura, Y.; Proske, F. N.; Puri, M. L.: Central limit theorems for generalized set-valued random variables. J. math. Anal. appl. 285, 250-263 (2003) · Zbl 1029.60022
[32] López-Díaz, M.; Gil, M. A.: Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications. J. statist. Plann. inference 74, 11-29 (1998) · Zbl 0962.62005
[33] M.A. Lubiano, Variation measures for imprecise random elements, Ph.D. Thesis, Universidad de Oviedo, 1999.
[34] Lubiano, M. A.; Gil, M. A.; López-Díaz, M.; López, M. T.: The $\lambda $→-mean squared dispersion associated with a fuzzy random variable. Fuzzy sets and systems 111, 307-317 (2000) · Zbl 0973.60005
[35] Molchanov, I.: On strong laws of large numbers for random upper semicontinuous functions. J. math. Anal. appl. 235, 349-355 (1999) · Zbl 0959.60003
[36] M. Montenegro, Estadística con Datos Imprecisos Basada en una Métrica Generalizada, Ph.D. Thesis, Universidad de Oviedo, 2003.
[37] Montenegro, M.; Casals, M. R.; Lubiano, M. A.; Gil, M. A.: Two-sample hypothesis tests of means of a fuzzy random variable. Inform. sci. 133, 89-100 (2001) · Zbl 1042.62012
[38] Montenegro, M.; Colubi, A.; Casals, M. R.; Gil, M. A.: Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable. Metrika 59, 31-49 (2004) · Zbl 1052.62048
[39] Montenegro, M.; González-Rodríguez, G.; Gil, M. A.; Colubi, A.; Casals, M. R.: Introduction to ANOVA with fuzzy random variables. Soft methodology and random information systems, 487-494 (2004) · Zbl 1055.62084
[40] Nguyen, H. T.: A note on the extension principle for fuzzy sets. J. math. Anal. appl. 64, 369-380 (1978) · Zbl 0377.04004
[41] Nguyen, H. T.; Kreinovich, V.; Wu, B.: Fuzzy/probability ∼ fractal/smooth. Internat. J. Uncert. fuzzy knowledge-based systems 7, 363-370 (1999) · Zbl 1113.68496
[42] Ogura, Y.; Li, S.: Separability for graph convergence of sequences of fuzzy-valued random variables. Fuzzy sets and systems 123, 19-27 (2001) · Zbl 0994.60035
[43] Ogura, Y.; Li, S.: Limit theorems for random fuzzy sets including large deviation principles. Soft methodology and random information systems, 32-44 (2004) · Zbl 1060.60029
[44] Okuda, T.; Tanaka, H.; Asai, K.: A formulation of fuzzy decision problems with fuzzy information using probability measures of fuzzy events. Inform. and control 38, 135-147 (1978) · Zbl 0401.94050
[45] Proske, F. N.; Puri, M. L.: Strong law of large numbers for Banach space valued fuzzy random variables. J. theor. Probab. 15, 543-551 (2002) · Zbl 1004.60029
[46] Puri, M. L.; Ralescu, D. A.: Différentielle d’une fonction floue. CR acad. Sci. Paris, sér. A 293, 237-239 (1981) · Zbl 0489.46038
[47] Puri, M. L.; Ralescu, D. A.: Differentials of fuzzy functions. J. math. Anal. appl. 91, 552-558 (1983) · Zbl 0528.54009
[48] Puri, M. L.; Ralescu, D. A.: The concept of normality for fuzzy random variables. Ann. probab. 11, 1373-1379 (1985) · Zbl 0583.60011
[49] Puri, M. L.; Ralescu, D. A.: Fuzzy random variables. J. math. Anal. appl. 114, 409-422 (1986) · Zbl 0592.60004
[50] Puri, M. L.; Ralescu, D. A.: Convergence theorem for fuzzy martingales. J. math. Anal. appl. 160, 107-122 (1991) · Zbl 0737.60005
[51] D.A. Ralescu, Fuzzy probabilities and their applications to statistical inference, in: B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh (Eds.), Advances in Intelligent Computing-IPMU’94, Lecture Notes in Computer Science, vol. 945, Springer, Paris, 1995, pp. 217 -- 222.
[52] Rodríguez-Muñiz, L.; López-Díaz, M.; Gil, M. A.: The s-differentiability of a fuzzy-valued mapping. Inform. sci. 151, 283-299 (2003) · Zbl 1027.26028
[53] Rodríguez-Muñiz, L.; López-Díaz, M.; Gil, M. A.: Differentiating random upper semicontinuous functions under the integral sign. Test 12, 241-258 (2003) · Zbl 1049.60013
[54] Rodríguez-Muñiz, L.; López-Díaz, M.; Gil, M. A.: Solving influence diagrams with fuzzy chance and value nodes. European J. Oper. res. 167, 444-460 (2005) · Zbl 1075.90040
[55] Román-Flores, H.; Rojas-Medar, M. A.: Differentiability of fuzzy-valued mapping. Rev. mat. Estatist 16, 223-239 (1998) · Zbl 0936.46056
[56] Stojaković, M.: Fuzzy random variables, expectation, and martingales. J. math. Anal. appl. 184, 594-606 (1994) · Zbl 0808.60005
[57] Terán, P.: Cones and decomposition of sub- and supermartingales. Fuzzy sets and systems 147, 465-474 (2004) · Zbl 1055.60039
[58] Terán, P.: An embedding theorem for convex fuzzy sets. Fuzzy sets and systems 152, 191-208 (2005) · Zbl 1087.46050
[59] P. Terán, A large deviation principle for random upper semicontinuous functions, Proc. Amer. Math. Soc. 134 (2006) 571 -- 580. · Zbl 1083.60020
[60] P. Terán, On Borel measurability and large deviations for fuzzy random variables, Fuzzy Sets and Systems, (2006) (in this issue). · Zbl 1109.60009
[61] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part 1. Inform. Sci. 8 (1975) 199 -- 249; Part 2. Inform. Sci. 8 (1975) 301 -- 353; Part 3. Inform. Sci. 9 (1975) 43 -- 80. · Zbl 0397.68071
[62] Zadeh, L. A.: Is probability theory sufficient for dealing with uncertainty in AI: A negative view. Mach intell. Pattern recognition 4, 103-116 (1986)