Document Type : Research Paper


1 Department of Statistics, Shahid Chamran University of Ahvaz, Ahvaz 83151-61355, Iran

2 School of Engineering Science, College of Engineering, University of Tehran, Tehran, P.O. Box 11365-4563, Iran


A novel approach is proposed for the problem of testing statistical hypotheses about the fuzzy mean of a fuzzy random variable.
The concept of the (uniformly) most powerful test is extended to the (uniformly) most powerful fuzzy-valued test in which the test function is a fuzzy set representing the degrees of rejection and acceptance of the hypothesis of interest.
For this purpose, the concepts of fuzzy test statistic and fuzzy critical value have been defined using the $\alpha$ cuts (levels) of the fuzzy observations and fuzzy parameter.
In order to make a decision as a fuzzy test, a well-known method is employed to compare the observed fuzzy test statistic and the fuzzy critical value.
In this work, we focus on the case in which the fuzzy data are observations of a normal fuzzy random variable.
The proposed approach is general so that it can be applied to other kinds of fuzzy random variables as well.
Numerical examples, including a lifetime testing problem, are provided to illustrate the proposed optimal tests.


[1] B. F. Arnold, An approach to fuzzy hypothesis testing, Metrika, 44 (1996), 119–126.
[2] B. F. Arnold, Testing fuzzy hypothesis with crisp data, Fuzzy Sets Syst., 9 (1998), 323–333.
[3] J. Behboodian and A. Mohammadpour, Using fuzzy knowledge of a nuisance parameter for
hypothesis testing, Iranian J. Sci. Tech. (Sciences), 29(3) (2005), 433–454.
[4] 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 Sa´a and B. Sinova, Random fuzzy sets: a mathematical tool to develop
statistical fuzzy data analysis, Iranian Journal of Fuzzy Systems, 10(2) (2013), 1–28.
[5] 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, A distance-based statistical analysis of fuzzy number
valued data, Int. J. Approx. Reason., 55 (2014), 1601–1605.
[6] J. Chachi and S. M. Taheri, Fuzzy confidence intervals for mean of Gaussian fuzzy random
variables, Expert Syst. Appl., 38 (2011), 5240–5244.
[7] J. Chachi, S. M. Taheri and R. Viertl, Testing statistical hypotheses based on fuzzy confidence
intervals, Austrian J. Stat., 41 (2012), 267–286.
[8] S. De La Rosa de Sa´a, M. ´A. Gil, G. Gonz´alez-Rodr´ıguez, M. T. L´opez and M. A. Lubiano,
Fuzzy rating scale-based questionnaires and their statistical analysis, IEEE Tran. Fuzzy Syst.,
23(1) (2015), 111–126.
[9] P. Filzmoser and R. Viertl, Testing hypotheses with fuzzy data: the fuzzy p-value, Metrika,
59 (2004), 21–29.
[10] M. ´A. Gil, M. Montenegro, G. Gonz´alez-Rodr´ıguez, A. Colubi M. R. Casals, Bootstrap approach
to the multi-sample test of means with imprecise data, Comput. Stat. Data Anal., 51
(2006), 148–162.
[11] G. Gonz´alez-Rodr´ıguez, M. Montenegro, A. Colubi and M. ´A. Gil, Bootstrap techniques and
fuzzy random variables: Synergy in hypothesis testing with fuzzy data, Fuzzy Sets Syst., 157
(2006), 2608–2613.
[12] P. Grzegorzewski, Statistical inference about the median from vague data, Control and Cybernetics,
27 (1998), 447–464.
[13] P. Grzegorzewski, Testing statistical hypotheses with vague data, Fuzzy Sets Syst., 112 (2000),
[14] P. Grzegorzewski, K-Sample median test for vague data, Int. J. Intell. Syst., 24 (2009),
[15] O. Hryniewicz, Goodman-Kruskal measure of dependence for fuzzy ordered categorical data,
Comput. Stat. Data Anal., 51 (2006), 323–334.
[16] O. Hryniewicz, Possibilistic decisions and fuzzy statistical tests, Fuzzy Sets Syst., 157 (2006),
[17] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall,
NJ, 1995.
[18] R. Kruse and K. D. Meyer, Statistics with Vague Data, Riedel Publishing, NY, 1987.
[19] E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, 3th ed., Springer, 2005.
[20] M. A. Lubiano, S. De La Rosa De S´aa, M. Montenegro, B. Sinova and M. ´A. Gil, Descriptive
analysis of responses to items in questionnaires. Why not using a fuzzy rating scale?, Inf.
Sci., 360 (2016), 131–148.
[21] M. A. Lubiano, M. Montenegro, B. Sinova, S. De La Rosa De S´aa and M.´A. Gil, Hypothesis
testing for means in connection with fuzzy rating scale-based data: algorithms and applications,
Europ. J. Oper. Res., 251 (2016), 918–929.
[22] M. A. Lubiano, A. Salas and M. ´A. Gil, A hypothesis testing-based discussion on the sensitivity
of means of fuzzy data with respect to data shape, Fuzzy Sets Syst., 328 (2017),
[23] M. Montenegro, M. R. Casals, M. A. Lubiano and M. ´A. Gil, Two-sample hypothesis tests of
means of a fuzzy random variable, Inf. Sci., 133 (2001), 89–100.
[24] 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, 59 (2004), 31–49.
[25] A. Parchami, S. M. Taheri and M. Mashinchi, Fuzzy p-value in testing fuzzy hypotheses with
crisp data, Stat. Papers, 51 (2010), 209–226.
[26] A. Parchami, S. M. Taheri, B. Sadeghpour Gildeh and M. Mashinchi, A simple but efficient
approach for testing fuzzy hypotheses, J. Uncertainty Anal. Appl., 4(2) (2016), 1–16.
[27] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
[28] P. Royston, An extension of Shapiro and Wilk’s W test for normality to large samples, Appl.
Stat., 31 (1982), 115–124.
[29] S. M. Taheri, Trends in fuzzy statistics, Austrian J. Stat., 32 (2003), 239–257.
[30] S. M. Taheri and M. Arefi, Testing fuzzy hypotheses based on fuzzy test statistic, Soft Computing,
13 (2009), 617–625.
[31] S. M. Taheri and J. Behboodian, Neyman-Pearson Lemma for fuzzy hypothesis testing,
Metrika, 49 (1999), 3–17.
[32] S. M. Taheri and J. Behboodian, A Bayesian approach to fuzzy hypotheses testing, Fuzzy
Sets Syst., 123 (2001), 39–48.
[33] S. M. Taheri and G. Hesamian, Goodman-Kruskal measure of association for fuzzycategorized
variables, Kybernetika, 47 (2011), 110–122.
[34] S. M. Taheri and G. Hesamian, A generalization of the Wilcoxon signed-rank test and its
applications, Stat Papers, 54 (2013), 457–470.
[35] H. Torabi, J. Behboodian and S. M. Taheri, Neyman-Pearson Lemma for fuzzy hypothesis
testing with vague data, Metrika, 64 (2006), 289–304.
[36] W. Trutschnig, M. A. Lubiano and J. Lastra, SAFD: An R package for statistical analysis
of fuzzy data, in: Borgelt C., Gil M., Sousa J., Verleysen M. (eds.), Towards Advanced Data 
Analysis by Combining Soft Computing and Statistics. Heidelberg, Germany: Springer, 285
(2013), 107-118.
[37] R. Viertl, Univariate statistical analysis with fuzzy data, Comput. Stat. Data Anal., 51
(2006), 133–147.
[38] R. Viertl, Statistical Methods for Fuzzy Data, John Wiley and Sons, Chichester, 2011.
[39] X. Wang and E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II),
Fuzzy Sets Syst., 118 (2001), 387–405.
[40] H. C. Wu, Statistical hypotheses testing for fuzzy data, Inf. Sci., 175 (2005), 30–57.
[41] H. C. Wu, Statistical confidence intervals for fuzzy data, Expert Syst. Appl., 36 (2009),
[42] S. Yosefi, M. Arefi and M. G. Akbari, A new approach for testing fuzzy hypotheses based on
likelihood ratio statistic, Stat Papers, 57 (2016), 665–688.
[43] Y. Yuan, Criteria for evaluating fuzzy ranking methods, Fuzzy Sets Syst., 43 (1991), 139–157.
[44] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353.