# TESTING STATISTICAL HYPOTHESES UNDER FUZZY DATA AND BASED ON A NEW SIGNED DISTANCE

Document Type : Research Paper

Author

Department of Statistics, Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran

Abstract

This paper deals with the problem of testing statistical
hypotheses when the available data are fuzzy. In this approach, we
first obtain a fuzzy test statistic based on fuzzy data, and then,
based on a new signed distance between fuzzy numbers, we introduce
a new decision rule to accept/reject the hypothesis of interest.
The proposed approach is investigated for two cases: the case
without nuisance parameters and the case with nuisance parameters.
This method is employed to test the hypotheses for the mean of a
normal distribution with known/unknown variance, the variance of a
normal distribution, the difference of means of two normal
distributions with known/unknown variances, and the ratio of
variances of two normal distributions.

Keywords

#### References

[1] M. Gh. Akbari and A. Rezaei, Bootstrap testing fuzzy hypotheses and observations on fuzzy
statistic, Expert Systems with Applications, 37 (2010), 5782{5787.
[2] T. Allahviranloo, S. Abbasbandy, and R. Saneifard, A method for ranking of fuzzy numbers
using new weighed distance, Mathematical and Computational Applications, 16 (2011), 359{
369.
[3] M. Arefi and S. M. Taheri, A fuzzy-based approach to testing statistical hypotheses, Interna-
tional Journal of Intelligent Technologies and Applied Statistics, 4(1) (2011), 109{132.
[4] M. Arefi and S. M. Taheri, Testing fuzzy hypotheses using fuzzy data based on fuzzy test
statistic, Journal of Uncertain Systems, 5(1) (2011), 45{61.
[5] M. Arefi and S. M. Taheri, A new approach for testing fuzzy hypotheses based on fuzzy data,
International Journal of Computational Intelligence Systems, 6(2) (2013), 318{327.
[6] B. F. Arnold, An approach to fuzzy hypothesis testing, Metrika, 44 (1996), 119{126.
[7] B. F. Arnold, Testing fuzzy hypotheses with crisp data, Fuzzy Sets and Systems 94 (1998),
323{333.
[8] C. Bertoluzza, N. Corall, and A. Salas, On a new class of dictances between fuzzy numbers,
Mathware & Soft Computing, 2 (1995), 71{84.
[9] P. P. Bonissone, Soft computing: the convergence of emerging reasoning technologies, Soft
Computing, 1 (1997), 6{18.
[10] J. J. Buckley, Fuzzy statistics: hypothesis testing, Soft Computing, 9 (2004), 512{518.
[11] M. R. Casals, and M. A. Gil, A note on the operativeness of Neyman-Pearson tests with
fuzzy information, Fuzzy Sets and Systems, 30 (1989), 215{220.
[12] M. R. Casals, M. A. Gil, and P. Gil, The fuzzy decision problem: an approach to the prob-
lem of testing statistical hypotheses with fuzzy information, European Journal of Operation
Research, 27 (1986), 371{382.
[13] J. Chachi, S. M. Taheri, and R. Viertl, Testing statistical hypotheses based on fuzzy confi dence
intervals, Austrian Journal of Statistics, 41(4)(2012), 267{286.
[14] S. H. Chen and CH Hsieh, Representation, ranking, distance, and similarity of L-R type fuzzy
number and application, Australian Journal of Intelligent Information Processing Systems,
6(4) (2000), 217{229.
[15] I. Couso, L. Sanchez, Defuzzifi cation of fuzzy p-values, In: Soft Methods for Handling Vari-
ability and Imprecision, D. Dubois et al. (Eds.), Advances in Soft Computing, Springer,
(2008), 126{132.
[16] M. Delgado, J. L. Verdegay, and M. A. Vila, Testing fuzzy hypotheses: A Bayesian approach,
In: Approximate Reasoningin Expert Systems, M. M. Gupta, A. Kandel, W. Bandlery, J. B.
Kiszka (Eds.), North-Holland, (1985), 307{316.
[17] J. G. Dijkman, H. Van Haeringen and S. J. De Lange, Fuzzy numbers, Journal of Mathemat-
ical Analysis and Applications, 92(2) (1983), 301{341.
[18] P. Filzmoser and R. Viertl, Testing hypotheses with fuzzy data: the fuzzy p-value, Metrika,
59 (2004), 21{29.
[19] P. Grzegorzewski, Testing fuzzy hypotheses with vague data, In: Statistical Modeling, Analysis
and Management of Fuzzy Data, C. Bertoluzza et al. (Eds.), Springer, Heidelberg, (2002),
213{225.
[20] P. Grzegorzewski, Testing statistical hypotheses with vague data, Fuzzy Sets and Systems,
112 (2000), 501{510.
[21] C. Kahraman, C. E. Bozdag, D. Ruan and A. F. Ozok, Fuzzy sets approaches to statistical
parametric and nonparametric tests, International Journal of Intelligent Systems, 19 (2004),
1{19.
[22] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentic-Hall,
Englewood Cliffs, NJ, 1995.
[23] R. Kruse and K. D. Meyer, Statistics with Vague Data, D. Reidel Publishing Company,
Dortrecht, Boston, 1987.
[24] X. Li and B. Liu, On distance between fuzzy variables, Journal of Intelligent and Fuzzy
Systems, 19(3) (2008), 197{204.
[25] J. Liu, L. Martinez, H. Wang, R. M. Rodriguez and V. Novozhilov, Computing with words in
risk assessment, International Journal of Computational Intelligence Systems, 3(4) (2010),
396{419.
[26] L. Magdalena, What is Soft Computing? Revisiting possible answers, International Journal
of Computational Intelligence Systems, 3(2) (2010), 148{159.
[27] A. Parchami, S. M. Taheri and M. Mashinchi, Fuzzy p-value in testing fuzzy hypotheses with
crisp data, Statistical Papers, 51 (2010), 209{226.
[28] A. Parchami, S. M. Taheri and M. Mashinchi, Testing fuzzy hypotheses based on vague ob-
servations: a p-value approach, Statistical Papers, 53(2) (2012), 469{484.
[29] D. Richards, Advanced Mathematical Methods with Maple, Cambridge University Press, New
York, 2002.
[30] R. Seising, What is Soft Computing? - Bridging gaps for 21st Century science!, International
Journal of Computational Intelligence Systems, 3(2) (2010), 160{175.
[31] S. M. Taheri Trends in fuzzy statistics, Austrian Journal of Statistics, 32 (2003), 239{257.
[32] S. M. Taheri and M. Arefi , Testing fuzzy hypotheses based on fuzzy test statistic, Soft Com-
puting, 13 (2009), 617{625.
[33] S. M. Taheri and J. Behboodian, Neyman-Pearson Lemma for fuzzy hypothesis testing,
Metrika, 49 (1999), 3{17.
[34] S. M. Taheri and J. Behboodian, A Bayesian approach to fuzzy hypotheses testing, Fuzzy
Sets and Systems, 123 (2001), 39{48.
[35] S. M. Taheri and J. Behboodian, On Bayesian approach to fuzzy testing hypothesis with fuzzy
data, Italian Journal of Pure and Applied Mathematics, 19 (2006), 139{154.
[36] S. M. Taheri and G. Hesamian A generalization of the Wilcoxon signed-rank test and its
applications, Statistical Papers, 54(2) (2013), 457{470.
[37] S. M. Taheri and G. Hesamian, Goodman-Kruskal measure of association for fuzzy-
categorized variables, Kybernetika, 47(1) (2011), 110{122.
[38] H. Torabi, J. Behboodian, and S. M. Taheri Neyman-Pearson Lemma for fuzzy hypotheses
testing with vague data, Metrika, 64 (2006), 289{304.
[39] W. Trutschnig, G. Gonzalez-Rodriguez, A. Colubi, and M. A. Gil, A new family of metrics for
compact, convex (fuzzy) sets based on a generalized concept of mid and spread, Information
Sciences, 179 (2009), 3964{3972.
[40] R. Viertl, Fuzzy models for precision measurements, Mathematics and Computers in Simu-
lation, 79 (2008), 874{878.
[41] R. Viertl, Statistical Methods for Fuzzy Data, J. Wiley, Chichester, 2011.
[42] J. S. Yao and K. M. Wu, Ranking fuzzy numbers based on decomposition principle and signed
distance, Fuzzy Sets and Systems, 116 (2000), 275{288.
[43] Sh. Yosefi , M. Arefi , and M. Gh. Akbari, A new approach for testing fuzzy hypotheses based
on likelihood ratio statistic, Statistical Papers, 57 (2016), 665{688.
[44] L. A. Zadeh, From computing with numbers to computing with words - from manipulation of
measurements to manipulation of perceptions, IEEE Transactions on Circuits and Systems-1:
Fundamental Theory and Applications, 46(1) (1999), 105{119.
[45] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.