×

The median of a random fuzzy number. The 1-norm distance approach. (English) Zbl 1260.60011

In classical statistics, it is well known that for estimation of the central tendency of a distribution the sample median is a much more robust statistics than the sample mean.
In the present paper, the authors find this (non surprising) property also for random fuzzy numbers (rfn). They define the median of a rfn as a fuzzy number which minimizes a suitable expected 1-norm distance with respect to the rfn and compare empirically and theoretically its robustness with that of Aumann expectation.

MSC:

60A86 Fuzzy probability
62F86 Parametric inference and fuzziness

Software:

SAFD
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bouchon-Meunier, B.; Kosheleva, O.; Kreinovich, V.; Nguyen, H.T., Fuzzy numbers are the only fuzzy sets that keep invertible operations invertible, Fuzzy sets syst., 91, 155-163, (1997) · Zbl 0920.04006
[2] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, () · Zbl 0346.46038
[3] Colubi, A.; Domínguez-Menchero, J.S.; López-Díaz, M.; Ralescu, D.A., On the formalization of fuzzy random variables, Inf. sci., 133, 3-6, (2001) · Zbl 0988.28008
[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. am. math. soc., 130, 3237-3242, (2002) · Zbl 1005.28003
[5] Colubi, A.; López-Díaz, M.; Domínguez-Menchero, J.S.; Gil, M.A., A generalized strong law of large numbers, Probab. theory relat. fields, 114, 401-417, (1999) · Zbl 0933.60023
[6] Couso, I.; Sánchez, L., The behavioral meaning of the Median, (), 115-123
[7] Cuevas, A.; Febrero, M.; Fraiman, R., On the use of the bootstrap for estimating functions with functional data, Comput. stat. data anal., 51, 1063-1074, (2006) · Zbl 1157.62390
[8] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, Fuzzy sets syst., 100, 63-71, (1999)
[9] Donoho, D.L.; Huber, P.J., The notion of breakdown point, (), 157-184
[10] Dubois, D.; Prade, H., Systems of linear fuzzy constraints, Fuzzy sets syst., 3, 37-48, (1980) · Zbl 0425.94029
[11] Faraz, A.; Shapiro, A.F., An application of fuzzy random variables to control charts, Fuzzy sets syst., 161, 2684-2694, (2010) · Zbl 1206.93062
[12] Fraimann, R.; Muñiz, G., Trimmed means for functional data, Test, 10, 419-440, (2001) · Zbl 1016.62026
[13] Gervini, D., Robust functional estimation using the spatial Median and spherical principal components, Biometrika, 95, 587-600, (2008) · Zbl 1437.62469
[14] Gil, M.A.; Montenegro, M.; González-Rodríguez, G.; Colubi, A.; Casals, M.R., Bootstrap approach to the multi-sample test of means with imprecise data, Comput. stat. data anal., 51, 148-162, (2006) · Zbl 1157.62391
[15] González-Rodríguez, G.; Colubi, A.; Gil, M.A., A fuzzy representation of random variables: an operational tool in exploratory analysis and hypothesis testing, Comput. stat. data anal., 51, 163-176, (2006) · Zbl 1157.62303
[16] G. González-Rodríguez, A. Colubi, M.A. Gil, Fuzzy data treated as functional data. A one-way ANOVA test approach, Comput. Stat. Data Anal., in press (doi:10.1016/j.csda.2010.06.013).
[17] González-Rodríguez, G.; Montenegro, M.; Colubi, A.; Gil, M.A., Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data, Fuzzy sets syst., 157, 2608-2613, (2006) · Zbl 1119.62037
[18] Hu, H.-Y.; Lee, Y.-C.; Yen, T.-M., Service quality gaps analysis based on fuzzy linguistic SERVQUAL with a case study in hospital out-patient services, Tqm j., 22, 499-515, (2010)
[19] Hukuhara, M., Intégration des applications measurables dont la valeur est un compact convexe, Funkcial. ekvac., 10, 205-223, (1967) · Zbl 0161.24701
[20] Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, Proc. R. soc. London A, 407, 171-182, (1986) · Zbl 0605.60038
[21] Körner, R., An asymptotic \(\alpha\)-test for the expectation of random fuzzy variables, J. stat. plann. inference, 83, 331-346, (2000) · Zbl 0976.62013
[22] Körner, R.; Näther, W., On the variance of random fuzzy variables, (), 22-39
[23] Li, S.; Ogura, Y., Strong laws of large numbers for independent fuzzy set-valued random variables, Fuzzy sets syst., 157, 2569-2578, (2006) · Zbl 1104.60307
[24] Luenberger, K., Optimization by vector space methods, (1968), Wiley New York
[25] Molchanov, I., On strong laws of large numbers for random upper semicontinuous functions, J. math. anal. appl., 235, 349-355, (1999) · Zbl 0959.60003
[26] Montenegro, M.; Casals, M.R.; Lubiano, M.A.; Gil, M.A., Two-sample hypothesis tests of means of a fuzzy random variable, Inf. sci, 133, 89-100, (2001) · Zbl 1042.62012
[27] 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
[28] Phillis, Y.A.; Kouikoglou, V.S., Fuzzy measurement of sustainability, (2009), Nova Sci. Pub. New York
[29] Proske, F.N.; Puri, M.L., A strong law of large numbers for generalized random sets from the viewpoint of empirical processes, Proc. am. math. soc., 131, 2937-2944, (2003) · Zbl 1020.60002
[30] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 11, 1373-1379, (1985) · Zbl 0583.60011
[31] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[32] Ramík, J.; Římánek, J., Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy sets syst., 16, 123-138, (1985) · Zbl 0574.04005
[33] M. Smithson, Applications of fuzzy set concepts to behavioral sciences, Math. Soc. Sci. 2 (1982) 257-274 (Inf. Sci. 179, 3964-3972). · Zbl 0483.92019
[34] Trutschnig, W.; González-Rodríguez, G.; Colubi, A.; Gil, M.A., A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread, Inf. sci., 179, 3964-3972, (2009) · Zbl 1181.62016
[35] W. Trutschnig, M.A. Lubiano, SAFD: Statistical Analysis of Fuzzy Data (R package) \(\langle\)http://bellman.ciencias.uniovi.es/SMIRE/SAFDpackage.html⟩, 2010. · Zbl 1348.62007
[36] Valvis, E., A new linear ordering of fuzzy numbers on subsets of \(\mathcal{F}(\mathbb{R})\), Fuzzy optim. decis. making, 8, 141-163, (2009) · Zbl 1179.03053
[37] Yager, R.R., A procedure for ordering fuzzy subsets of the unit interval, Inf. sci., 24, 143-161, (1981) · Zbl 0459.04004
[38] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part 1, Inf. Sci. 8 (1975) 199-249; Part 2, Inf. Sci. 8, 301-353; Part 3, Inf. Sci. 9, 43-80. · Zbl 0397.68071
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.