Bootstrap Approach to the One-Sample and Two- Sample Test of Variances of a Fuzzy Random Variable

  • Jalal Chachi Semnan University
Keywords: −pessimistic, Bootstrap, Fuzzy random variable, Testing hypotheses


The aim of this paper is to present in a concise and integrated way of the bootstrap approach to statistical testing of hypotheses about the variance of  fuzzy random variable. In this approach, first a notion of fuzzy random variables is recalled. Then, we will consider hypothesis-tests for the (crisp-valued)  variance   of   fuzzy data in a population. For this purpose, the $\alpha$-pessimistic values of the imprecise observations are used for defining a new notion of distance measure  between  fuzzy data, which is then used to make a procedure for testing the statistical hypotheses. Based on this argument,  the application of bootstrap techniques in dealing with these testing problems will be introduced. The procedure develops a non-parametric approach to testing statistical hypotheses based on one-sample and two-sample fuzzy data.

Author Biography

Jalal Chachi, Semnan University
Department of Mathemathics, Statistics and Computer Sciences,Semnan University,Semnan,Iran


M. G. Akbari, and A. Rezaei, Bootstrap testing fuzzy hypotheses and observations on fuzzy statistic, Expert Systems with Applications, vol. 37, pp. 5782–5787, 2010.

K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag, Heidelberg, 1999.

R. J. Auman, Integrals of set-valued functions, Journal of Mathematical Analysis and Applications, vol. 12, pp. 1–12, 1965.

P. Billingsley, Probability and Measure, 3rd ed, Wiley, New York, 1995.

A., Blanco-Fern´andez, M.R., Casals, A., Colubi, N., Corral, M., Garc´ıa-B´arzana, M.A., Gil, G., Gonz´alez-Rodr´ıguez, M.T., L´opez, M.A., Lubiano, M., Montenegro, A.B., Ramos-Guajardo, S. De La Rosa De S´aa, and B. Sinova, Random fuzzy sets: a mathematical tool to develop statistical fuzzy data analysis, Iranian Journal of Fuzzy Systems, vol. 10, pp. 1–28, 2013.

J. Chachi, On distribution characteristics of a fuzzy random variable, Austrian Journal of Statistics, 2017, In Press.

J. Chachi, and S.M. Taheri, Optimal statistical tests based on fuzzy random variables, Iranian Journal of Fuzzy Systems, 2017, In Press.

B. Efron, and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, London, 1993.

M. Frechet, Les elements aleatoires de natures quelconque dans une space distancie, Ann. Inst. H. Poincare, vol. 10, no. 4, pp. 215–310, 1948. (In Ferench).

W. L. Gau, and D. J. Buehrer, Vague sets, IEEE Transactions on Systems Man and Cybernetics, vol. 23, pp. 610–614, 1993.

M.A . Gil, M. Montenegro, G. Gonzalez-Rodrıguez, A. Colubi, and M. R. Casals, Bootstrap approach to the multi-sample test of means with imprecise data, Computational Statistics and Data Analysis, vol. 51, pp. 148–162, 2006.

M.A. Gil, M. Lopez-Dıaz, and D. A. Ralescu, Overview on the development of fuzzy random variables, Fuzzy Sets and Systems,vol. 157, pp. 2546–2557, 2006.

G. Gonzalez-Rodrıguez, M. Montenegro, A. Colubi, and M. A. Gil, Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data, Fuzzy Sets and Systems, vol. 157, pp. 2608–2613, 2006.

M. B. Gorzalzany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems,vol. 21, pp. 1–17, 1987.

G. Hesamian, and J. Chachi, Two-sample Kolmogorov-Smirnov fuzzy test for fuzzy random variables, Statistical Papers, vol. 56, pp.61–82, 2013.

R. Korner, On the variance of random fuzzy variables, Fuzzy Sets and Systems, vol. 92, no. 1, pp. 83–93, 1997.

V. Kratschmer, A unified approach to fuzzy random variables, Fuzzy Sets and Systems, vol. 123, pp. 1–9, 2001.

R. Kruse, and K. D. Meyer, Statistics with Vague Data, Reidel Publishing Company, Dordrecht, 1987.

H. Kwakernaak, Fuzzy random variables part I: Definitions and theorems, Information Sciences, vol. 19, pp. 1–15, 1978.

H. Kwakernaak, Fuzzy random variables part II: Algorithms and examples for the discrete case, Information Sciences, vol. 17, pp.253–278, 1979.

B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002.

B. Liu, Uncertainty Theory, 5th ed, Springer-Verlag, Berlin (URL:, 2016.

B. Liu, and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, vol.10, pp. 445–450, 2002.

Y. K. Liu, and B. Liu, Fuzzy random variables: A scalar expected value operator, Fuzzy Optimization and Decision Making, vol.2, pp. 143–160, 2003.

M. Montenegro, A. Colubi, M. R. Casals, and M.A´ . Gil, Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable, Metrika, vol. 59, pp. 31–49, 2004.

J. Peng, and B. Liu, Some properties of optimistic and pessimistic values of fuzzy, IEEE International Conference on Fuzzy Systems, vol. 2, pp. 745–750, 2004.

M. L. Puri, and D. A. Ralescu, The concept of normality for fuzzy random variables, The Annals of Probability, vol. 13, pp. 1373–1379, 1985.

M. L. Puri, and D. A. Ralescu, D.A., Fuzzy random variables, Journal of Mathematical Analysis and Applications, vol. 114, pp. 409–422, 1986.

A. B. Ramos-Guajardo, A. Colubi, G. Gonzalez-Rodrıguez, and M. A. Gil, One sample tests for a generalized Fr´echet variance of a fuzzy random variable, Metrika, vol. 71, no. 2, pp. 185–202, 2010.

A. B. Ramos-Guajardo, and M. A. Lubiano, K-sample tests for equality of variances of random fuzzy sets, Computational Statistics and Data Analysis, vol. 56, no. 4, pp. 956–966, 2012.

J. Shao, and D. Tu, The Jackknife and Bootstrap. Springer, NewYork, 1995.

S. M. Taheri, and G. Hesamian, A generalization of the Wilcoxon signed-rank test and its applications, Stat Papers, vol. 54, pp.457–470, 2013.

R. Viertl, Statistical Methods for Fuzzy Data, Wiley, Chichester, 2011.

R. A. Vitale, An alternate formulation of mean value for random geometric figures, J. Microscopy, vol. 151, no. 3, pp. 197–204,1998.

S. Yosefi, M. Arefi, and M. G. Akbari, A new approach for testing fuzzy hypotheses based on likelihood ratio statistic, Stat Papers, vol. 57, pp. 665–688, 2016.

L. A. Zadeh, Fuzzy sets, Information and Control, vol. 8, pp. 338–353, 1965.

Z. Zainali, M. G. Akbari, and H. Alizadeh Noughabi, Intuitionistic fuzzy random variable and testing hypothesis about its variance, Soft Computing, vol. 19, no. 9, pp. 2681–2689, 2015.

H. J. Zimmermann, Fuzzy Set Theory and Its Applications, 4th ed., Kluwer Nihoff, Boston, 2001.

How to Cite
Chachi, J. (2017). Bootstrap Approach to the One-Sample and Two- Sample Test of Variances of a Fuzzy Random Variable. Statistics, Optimization & Information Computing, 5(3), 188-199.
Research Articles