A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing. (English) Zbl 1157.62303

Summary: A family of fuzzy representations of random variables is presented. Each representation transforms a real-valued random variable into a fuzzy-valued one. These representations can be chosen so that they lead to fuzzy random variables whose means capture different relevant information on the probability distribution of the original real-valued random variable. In this way, the means of the transformed fuzzy random variables can capture, for instance, immediate visual information about some key parameters, and even the whole information about the distribution of the original random variable. Representations capturing visual information on parameters of the original random variable may be considered for statistical descriptive/exploratory purposes. Representations for which the fuzzy mean characterizes the distribution of the original random variable will be mainly valuable to develop statistical inferences on this variable. Some interesting inferential applications for classical random variables based on the last fuzzy representations are commented, and an example illustrates one of them empirically and motivate future directions and discussions.


62-07 Data analysis (statistics) (MSC2010)
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[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] Colubi, A.; Domínguez-Menchero, J.S.; López-Díaz, M.; Gil, M.A., A generalized strong law of large numbers, Probab. theory related fields, 114, 401-417, (1999) · Zbl 0933.60023
[3] Colubi, A., Domínguez-Menchero, J.S., López-Díaz, M., Relescu, D.A., 2001. On the formalization of fuzzy random variables. Inform. Sci. 133, 3-6.
[4] Colubi, A.; Domínguez-Menchero, J.S.; López-Díaz, M.; Ralescu, D.A., A \(D_E [0, 1]\)-representation of random upper semicontinuous functions, Proc. amer. math. soc., 130, 3237-3242, (2002) · Zbl 1005.28003
[5] Gil, M.A., Montenegro, M., González-Rodríguez, G., Colubi, A., Casals, M.R., 2006. Bootstrap approach to the multisample test of means with imprecise data. Comput. Statist. Data Anal., this issue, doi: 10.1016/j.csda.2006.04.018.
[6] González-Rodríguez, G., Colubi, A., 2005. Empirical discussions on the power associated with the triangular fuzzification of binomial populations in two-sided tests. Abstracts IASC Third World Conference on Comp. Statistics & Data Anal. (http://www.csdassn.org/europe/CSDA2005/), pp. 70-71.
[7] González-Rodríguez, G., Montenegro, M., Colubi, A., Gil, M.A., 2006. Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data. Fuzzy Sets and Systems, in press.
[8] Klement, E.P.; Puri, M.L.; Ralescu, D.A., Law of large numbers and central limit theorems for fuzzy random variables, (), 525-529
[9] Körner, R., An asymptotic \(\alpha\)-test for the expectation of random fuzzy variables, J. statist. plann. inference, 83, 331-346, (2000) · Zbl 0976.62013
[10] Kruse, R.; Meyer, K.D., Statistics with vague data, (1987), Reidel Publ. Co Dordrecht · Zbl 0663.62010
[11] 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
[12] López-Díaz, M., Ralescu, D.A., 2006. Tools for fuzzy random variables: embeddings and measurabilities. Comput. Statist. Data Anal., in this issue.
[13] Molchanov, I., On strong laws of large numbers for random upper semicontinuous functions, J. math. anal. appl., 235, 349-355, (1999) · Zbl 0959.60003
[14] 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
[15] 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
[16] Montenegro, M., González-Rodríguez, G., Colubi, A., Gil, M.A., 2005. Bootstrap techniques: a valuable tool in statistical hypothesis testing about the means of fuzzy random variables. Proceedings of Joint EUSFLAT-LFA, 2005, pp. 599-604.
[17] Proske, F.N.; Puri, M.L., Strong law of large numbers for Banach space valued fuzzy random variables, J. theoret. probab., 15, 543-551, (2002) · Zbl 1004.60029
[18] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[19] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, part 1, Inform. sci., 8, 199-249, (1975), Part 2. Inform. Sci. 8, 301-353; Part 3, Inform. Sci. 9, 43-80 · Zbl 0397.68071
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