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Linear regression with random fuzzy variables: Extended classical estimates, best linear estimates, least squares estimates. (English) Zbl 0930.62072

Summary: This paper deals with problems which arise if for well justified statistical models like linear regression only fuzzy data are available. Three approaches are discussed: The first one is an application of Zadeh’s extension principle to optimal classical estimators. Here, the result is that they, in general, do not keep their optimality properties. The second one is the attempt to develop a certain kind of linear theory for fuzzy random variables w.r.t. extended addition and scalar multiplication. The main problem in this connection is that fuzzy sets, with these extended operations, do not constitute a linear space. Therefore, in the third approach, the least squares approximation principle for fuzzy data is investigated, which leads to most acceptable results.

MSC:

62J99 Linear inference, regression
62J05 Linear regression; mixed models
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References:

[1] Kwakernaak, H., Fuzzy random variables — I. definitions and theorems, Inform. sci., 15, 1-29, (1978) · Zbl 0438.60004
[2] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[3] Kruse, R.; Meyer, K.D., Statistics with vague data, (1987), Reidel Dordrecht, Boston · Zbl 0663.62010
[4] Tanaka, H., Fuzzy data analysis by possibility linear models, Fuzzy sets and systems, 24, 363-375, (1987) · Zbl 0633.93060
[5] Sakawa, M.; Yano, M., Fuzzy linear regression and its applications, (), 61-80, Physica-Verlag, Heidelberg · Zbl 0778.62060
[6] Kacprzyk, J.; Fedrizzi, M., Fuzzy regression analysis, (1992), Omnitech Press Warsaw, Physica-Verlag, Heidelberg
[7] Diamond, P., Fuzzy least squares, Inform. sci., 46, 141-157, (1988) · Zbl 0663.65150
[8] Diamond, P., Least squares and maximum likelihood regression for fuzzy linear models, (), 137-151, Physica-Verlag, Heidelberg · Zbl 0778.62059
[9] Bardossy, A.; Hagaman, R.; Duckstein, L.; Bogardi, I., Fuzzy least squares regression: theory and application, (), 183-193, Physica-Verlag, Heidelberg · Zbl 0825.62641
[10] Fre´chet, M., Lese´le´ments ale´atoires de natures quelconque dans une´space distancie´, Ann. inst. H. poincare´, 10, 215-310, (1948)
[11] Na¨ther, W., Linear statistical inference for random fuzzy data, Statistics, 29, 221-240, (1997)
[12] R. Ko¨rner, On the variance of fuzzy random variables, Fuzzy Sets and Systems 92/1 (accepted for publication).
[13] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton-New Jersey · Zbl 0229.90020
[14] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific Singapore · Zbl 0843.54041
[15] Luenberger, K., Optimization by vector space methods, (1968), Wiley New York
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.