Linear statistical inference for random fuzzy data. (English) Zbl 1030.62530

Summary: The paper present first steps towards linear statistical inference with random fuzzy data. In establishing best linear unbiased estimators (BLUE) or best linear predictors (BLP) it is necessary to consider a suitable notion of expectation and variance for random fuzzy variables. Using \(\alpha\)-cuts of the random fuzzy variables, the mathematical problem can be formulated within the theory of random closed sets (RCS). As a methodological guide we use the Fréchet-approach which leads for a given metric to an associated expectation and variance. In particular, we prove that the well known Aumann expectation is a Fréchet expectation and that the associated variance appears as a reasonable definition for a variance of RCS’s. this associated variance has the advantage that at least special fuzzy number data can formally be handled like Euclidean vectors. Due to the special addition and scalar multiplication these “vectors”, however, fail to build a linear space which leads to somewhat more complicated optimization procedures for BLUE and BLP. This is demonstrated for linear regression and for prediction of stationary processes. In special cases, however, the fuzzified version of the classical BLUE and BLB keep their optimality.


62J99 Linear inference, regression
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] DOI: 10.1016/0022-247X(65)90049-1 · Zbl 0163.06301
[2] Bandemer, H. and Gebhardt, A. ”Bayesian Fuzzy Kriging, submitted to Mathematical Geology”. · Zbl 1040.62088
[3] DOI: 10.1016/0020-0255(88)90047-3 · Zbl 0663.65150
[4] DOI: 10.1016/0165-0114(89)90121-8 · Zbl 0679.62088
[5] Diamond Ph., Metric Spaces of Fuzzy Sets: Theory and Application (1994)
[6] Dubois D., Fuzzy Sets and Systems: Theory and Applications (1980) · Zbl 0444.94049
[7] Fréchet M., Ann. Inst. H. Poincaré 10 pp 215– (1948)
[8] Kruse R., Statistics with rague data (1987)
[9] DOI: 10.1016/0020-0255(78)90019-1 · Zbl 0438.60004
[10] DOI: 10.1080/02331889008802262 · Zbl 0714.62063
[11] DOI: 10.1016/0022-247X(86)90093-4 · Zbl 0592.60004
[12] Stoyan D., Fractals, Random Shapes and Point Fields (1994) · Zbl 0828.62085
[13] DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606
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