Document Type : Research Paper


Department of Statistics and Probability Theory, Vienna University of Technology, Wien, Austria


In applications there occur different forms of uncertainty. The two
most important types are randomness (stochastic variability) and imprecision
(fuzziness). In modelling, the dominating concept to describe uncertainty is
using stochastic models which are based on probability. However, fuzziness
is not stochastic in nature and therefore it is not considered in probabilistic
Since many years the description and analysis of fuzziness is subject of intensive
research. These research activities do not only deal with the fuzziness of
observed data, but also with imprecision of informations. Especially methods
of standard statistical analysis were generalized to the situation of fuzzy observations.
The present paper contains an overview about of the presentation
of fuzzy information and the generalization of some basic classical statistical
concepts to the situation of fuzzy data.


[1] E.P. Klement, M.L. Puri, D.A. Ralescu, Law of large numbers and central limit theorem for
fuzzy random variables, Cybernetics and Systems Research 2, Proc. 7th Europ. Meet., Vienna
1984, 525-529 (1984).
[2] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Acad. Publ., Dordrecht, 2000 .
[3] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic - Theory and Applications, Prentice Hall, Upper
Saddle River, New Jersey, 1995 .
[4] R. Kruse, The strong law of large numbers for fuzzy random variables, Information Science,
Vol. 28 (1982) 233-241 .
[5] H. Kwakernaak, Fuzzy random variables - I. definition and theorems, Information Science,
Vol. 15 (1978) 1-29 .
[6] H. Kwakernaak, Fuzzy random variables - II. algorithms and examples, Information Science,
Vol. 17 (1979) 253-278 .
[7] B. M¨oller, W. Graf, M. Beer, Safety assessment of structures in view of fuzzy randomness,
Computers & Structures, Vol. 81 (2003).
[8] S. Niculescu, R. Viertl, Bernoulli’s Law of Large Numbers for Vague Data, Fuzzy Sets and
Systems, Vol. 50 (1992).
[9] M.L. Puri, D.A. Ralescu, Fuzzy Random Variables, Journal of Math. Anal. and Appl., Vol.
114 (1986) 409-422 .
[10] C. R¨omer, A. Kandel, Statistical tests for fuzzy data, Fuzzy Sets and Systems, Vol. 72 (1995).
[11] J. Sickert, M. Beer, W. Graf, B. M¨oller, Fuzzy probabilistic structural analysis considering
fuzzy random functions, in A. Kiureghian, S. Madanat, J. Pestana (Eds.), Applications of
Statistics and Probability in Civil Engineering, Milpress, Rotterdam, 2003 .
[12] S.M. Taheri, Trends in Fuzzy Statistics, Austrian Journal of Statistics, Vol. 32, No. 3 (2003)
239-257 .
[13] R. Viertl, Statistical Methods for Non-Precise Data, CRC Press, Boca Raton, Florida, 1996
[14] R. Viertl, On the description and analysis of measurements of continuous quantities, Kybernetika,
Vol. 38 (2002).
[15] P. Filzmoser, R. Viertl, Testing Hypotheses with Fuzzy Data, The Fuzzy p-value, to appear
in Metrika.
[16] R. Viertl, D. Hareter, Fuzzy Information and Imprecise Probability, to appear in ZAMM.
[17] W. Voß (Ed.), Taschenbuch der Statistik, Carl Hauser Verlag, M¨unchen, 2004.
[18] G. Wang, Y. Zhang, The theory of fuzzy stochastic processes, Fuzzy Sets and Systems, Vol.
51 (1992).