On the variance of fuzzy random variables. (English) Zbl 0936.60017

Frechét has defined an expectation of a random fuzzy variable (rfv) \(X\) with values in a metric space \((M,d)\) by that \(a^*\in M\) which minimizes \(Ed^2(X,a)\) and he has further defined \(\text{Var} X:=Ed^2(X,a^*)\). In the present paper it is shown that the Aumann-expectation of a rfv is Frechét w.r.t. the \(L^2\)-metric between support functions, associated with convex rfv’s. Then the Frechét-principle leads to an appropriate variance of rfv’s. The author discusses properties of that variance and presents special formulas for random \(LR\)-fuzzy-numbers. As application, best linear estimation and best linear prediction in linear regression models with fuzzy data and a strong law of large numbers w.r.t. the used \(L^2\)-metric is considered.


60E99 Distribution theory
60A99 Foundations of probability theory
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Artstein, Z.; Vitale, R.A., A strong law of large numbers for random compact sets, Ann. probab., 5, 879-882, (1975) · Zbl 0313.60012
[2] Diamond, Ph.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific Singapore, New Jersey, London, Hong Kong
[3] Fréchet, M., LES éléments aléatoires de natures quelconque dans un espace distancié, Ann. inst. H. Poincaré, 10, 215-310, (1948) · Zbl 0035.20802
[4] Körner, R., A variance of compact convex random sets, Statistics, (1995), submitted
[5] Kruse, R.; Meyer, K.D., Statistics with vague data, (1987), Reidel Dordrecht, Boston · Zbl 0663.62010
[6] Lyashenko, N.N., Limit theorems for sums of independent compact random subsets of Euclidean space, J. soviet. math., 20, 2187-2196, (1982) · Zbl 0489.60041
[7] Näther, W., Linear statistical inference with random fuzzy data, Statistics, (1995), submitted
[8] Rényi, A., Probability theory, (1970), Akadémiai Kiadó Budapest · Zbl 0206.18002
[9] Schneider, R., Convex bodies, the brunn-Minkowski theory, (1993), Cambridge Univ. Press Cambridge · Zbl 0798.52001
[10] Vitale, R.A., An alternate formulation of Mean value for random geometric figures, J. microscopy, 151, 197-204, (1988)
[11] Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606
[12] Ziezold, H., On expected figures in the plane, (), 105-110
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.