×

Estimation of a simple linear regression model for fuzzy random variables. (English) Zbl 1175.62073

Summary: A generalized simple linear regression statistical/probabilistic model where both input and output data can be fuzzy subsets of \(\mathbb R^p\) is dealt with. The regression model is based on a fuzzy-arithmetic approach and it considers the possibility of fuzzy-valued random errors. Specifically, the least-squares estimation problem in terms of a versatile metric is addressed. The solutions are established in terms of the moments of the involved random elements by employing the concept of support function of a fuzzy set. Some considerations concerning the applicability of the model are made.

MSC:

62J05 Linear regression; mixed models
62J86 Fuzziness, and linear inference and regression
62F10 Point estimation
62F86 Parametric inference and fuzziness
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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
[2] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Ralescu, D. A., A \(D_E [0, 1]\)-representation of random upper semicontinuous functions, Proc. Amer. Math. Soc., 130, 3237-3242 (2002) · Zbl 1005.28003
[3] Colubi, A.; González-Rodríguez, G., Triangular fuzzification of random variables and power of distribution tests: empirical discussion, Comput. Statist. Data Anal., 51, 9, 4742-4750 (2007) · Zbl 1162.62342
[4] Coppi, R.; D’Urso, P.; Giordani, P.; Santoro, A., Least squares estimation of a linear regression model with lr fuzzy response, Comput. Statist. Data Anal., 51, 267-286 (2006) · Zbl 1157.62460
[5] Diamond, P., Fuzzy least squares, Inform. Sci., 46, 141-157 (1988) · Zbl 0663.65150
[6] Diamond, P., Least squares fitting of compact set-valued data, J. Math. Anal. Appl., 14, 531-544 (1990)
[7] Diamond, P., Least squares and maximum likelihood regression for fuzzy linear models, (Kacprzyk, J.; Fedrizzi, M., Fuzzy Regression Analysis (1992), Omnitech Press, Warsaw and Physica-Verlag: Omnitech Press, Warsaw and Physica-Verlag Heidelberg), 137-151 · Zbl 0778.62059
[8] Diamond, P.; Kloeden, P., Metric Spaces of Fuzzy Sets: Theory and Applications (1994), World Scientific: World Scientific Singapore · Zbl 0873.54019
[9] Diamond, P.; Körner, R., Extended fuzzy linear models and least squares estimates, Comput. Math. Appl., 33, 15-32 (1997) · Zbl 0936.62073
[10] Gallant, A. R.; Gerig, T. M., Computations for constrained linear models, J. Econometrics, 12, 59-89 (1980) · Zbl 0432.62043
[11] García, D.; Lubiano, M. A.; Alonso, M. C., Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations, Inform. Sci., 138, 165-184 (2001) · Zbl 1003.62005
[12] Gil, M.; González-Rodríguez, G.; Colubi, A.; Montenegro, M., Testing linear independence in linear models with interval-valued data, Comput. Statist. Data Anal., 51, 3002-3015 (2007) · Zbl 1161.62358
[13] Gil, M.; López-García, M. T.; Lubiano, M. A.; Montenegro, M., Regression and correlation analyses of a linear relation between random intervals, Test, 10, 1, 183-201 (2001) · Zbl 0981.62062
[14] Gil, M.; Lubiano, M. A.; Montenegro, M.; López-García, M. T., Least squares fitting of an affine function and strength of association for interval-valued data, Metrika, 56, 97-111 (2002) · Zbl 1433.60004
[15] Gil, M.; Montenegro, M.; González-Rodríguez, G.; Colubi, A.; Casals, M., Bootstrap approach to the multi-sample test of means with imprecise data, Comput. Statist. Data Anal., 51, 1, 148-162 (2006) · Zbl 1157.62391
[16] González-Rodríguez, G.; Blanco, A.; Corral, N.; Colubi, A., Least squares estimation of linear regression models for convex compact random sets, Adv. Data Anal. Class., 1, 67-81 (2007) · Zbl 1131.62058
[17] González-Rodríguez, G.; Colubi, A.; Coppi, R.; Giordani, P., On the estimation of linear models with interval-valued data, (Proc. 17th Conf. of IASC-ERS (COMPSTAT’2006) (2006)), 697-704
[18] Hukuhara, M., Intégration des applications measurable dont la valeur est un compact convexe, Funkcial. Ekvac., 10, 205-223 (1967) · Zbl 0161.24701
[19] Körner, R.; Näther, W., Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates, Inform. Sci., 109, 95-118 (1998) · Zbl 0930.62072
[20] Körner, R.; Näther, W., On the variance of random fuzzy variables, (Bertoluzza, C.; Gil, M.; Ralescu, D., Statistical Modeling, Analysis and Management of Fuzzy Data (2002), Physica-Verlag: Physica-Verlag Heidelberg), 22-39
[21] Krätschmer, V., Least squares estimation in linear regression models with vague concepts, (López-Díaz, M.; Gil, M. A.; Grzegorzewski, P.; Hryniewicz, O.; Lawry, J., Soft Methodology and Random Information Systems (2004), Springer: Springer Heidelberg), 407-414 · Zbl 1055.62081
[22] Krätschmer, V., Limit distributions of least squares estimators in linear regression models with vague concepts, J. Multivariate Anal., 97, 1044-1069 (2006) · Zbl 1119.62014
[23] Krätschmer, V., Strong consistency of least-squares estimation in linear regression models with vague concepts, J. Multivariate Anal., 97, 633-654 (2006) · Zbl 1085.62025
[24] Liew, C. K., Inequality constrained least-squares estimation, J. Amer. Statist. Assoc., 71, 746-751 (1976) · Zbl 0342.62037
[25] Lubiano, M. A.; Gil, M. A.; López-Díaz, M.; López-García, M., The \(\overrightarrow{\lambda} \)-mean squared dispersion associated with a fuzzy random variable, Fuzzy Sets and Systems, 111, 307-317 (2000) · Zbl 0973.60005
[26] 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
[27] Montenegro, M.; González-Rodríguez, G.; Gil, M. A.; Colubi, A.; Casals, M. R., Introduction to ANOVA with fuzzy random variables, (López-Díaz, M.; Gil, M. A.; Grzegorzewski, P.; Hryniewicz, O.; Lawry, J., Soft Methodology and Random Information Systems (2004), Springer: Springer Berlin), 487-494 · Zbl 1055.62084
[28] Näther, W., On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data, Metrika, 51, 3, 201-221 (2000) · Zbl 1093.62557
[29] Näther, W., Regression with fuzzy random data, Comput. Statist. Data Anal., 51, 235-252 (2006) · Zbl 1157.62463
[30] Puri, M. L.; Ralescu, D. A., Différentielle d’une fonction floue, C. R. Acad. Sci. Paris Sér. A, 293, 237-239 (1981) · Zbl 0489.46038
[31] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl., 114, 409-422 (1986) · Zbl 0592.60004
[32] Wünsche, A.; Näther, W., Least-squares fuzzy regression with fuzzy random variables, Fuzzy Sets and Systems, 130, 43-50 (2002) · Zbl 1010.62067
[33] Zadeh, L., The concept of a linguistic variable and its application to approximate reasoning, I, Inform. Sci., 8, 199-249 (1975) · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.