# zbMATH — the first resource for mathematics

Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data. (English) Zbl 1119.62037
Summary: In previous studies we have stated that the well-known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables, when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one-dimensional Euclidean space. In this paper, we show that the one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). This extension is immediately developed by combining some tools in the literature, namely, bootstrap techniques on Banach spaces, a metric between fuzzy sets based on the support function, and an embedding of the space of fuzzy random variables into a Banach space which is based on the support function.

##### MSC:
 62G09 Nonparametric statistical resampling methods 62G10 Nonparametric hypothesis testing
Full Text:
##### References:
 [1] A.P. Araujo, E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1980. [2] Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301 [3] 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 [4] 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 syst., 10, 384-390, (2002) [5] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific Singapore · Zbl 0843.54041 [6] M.A. Gil, M., López-Díaz, D.A. Ralescu, Overview on the development of fuzzy random variables, Fuzzy Sets and Systems, 2006, in this issue. [7] M.A. Gil, M. Montenegro, G. González-Rodríguez, A. Colubi, M.R. Casals, Bootstrap approach to the classic oneway multi-sample test with imprecise data, Comp. Stat. Data Anal., 2006, in press. [8] Giné, E.; Zinn, J., Bootstrapping general empirical measures, Ann. probab., 18, 851-869, (1990) · Zbl 0706.62017 [9] Körner, R., An asymptotic $$\alpha$$-test for the expectation of random fuzzy variables, J. stat. plann. inference, 83, 331-346, (2000) · Zbl 0976.62013 [10] Körner, R.; Näther, W., On the variance of random fuzzy variables, (), 22-39 [11] M. Montenegro, M.R. Casals, M.A. Gil, Asymptotic comparison of two fuzzy expected values, Proc. JCIS 2000—Seventh FT&T Conference, 2000, pp. 150-153. [12] 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 [13] 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 [14] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004 [15] Zadeh, L.A.; Zadeh, L.A.; Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, Part 1. inform. sci., Part 2. inform. sci., Part 3. inform. sci., 9, 43-80, (1975) · Zbl 0404.68075
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.