Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications. (English) Zbl 0962.62005

Summary: Conditions are given to compute iterated expectations of fuzzy random variables irrespectively of the order of integration. To this purpose, some studies about the measurability and the integrable boundedness of the fuzzy expected value of a fuzzy random variable with respect to a regular conditional probability are first developed. The conclusions obtained are applied later to statistical problems involving fuzzy random variables, like those concerning some hierarchical models and mixture distributions (more precisely, some techniques to obtain fuzzy unbiased estimators and the Bayesian analysis of statistical problems).


62A01 Foundations and philosophical topics in statistics
62C10 Bayesian problems; characterization of Bayes procedures
62C99 Statistical decision theory
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
62H12 Estimation in multivariate analysis
62H15 Hypothesis testing in multivariate analysis
62H99 Multivariate analysis
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Ash, R., 1975. Real Analysis and Probability. Academic Press, New York.
[2] Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301
[3] Breiman, L., 1968. Probability. Addison-Wesley, Reading, MA. · Zbl 0174.48801
[4] Brown, L.D.; Purves, R., Measurable selection of extrema, Ann. statist., 1, 902-912, (1973) · Zbl 0265.28003
[5] Byrne, C., Remarks on the set-valued integrals of Debreu and Aumann, J. math. anal. appl., 78, 243-246, (1978) · Zbl 0373.28005
[6] Campos, L.M.; González, A., A subjective approach for ranking fuzzy numbers, Fuzzy sets and systems, 29, 145-153, (1989) · Zbl 0672.90001
[7] Casella, G., Berger, R.L., 1990. Statistical Inference. Wadsworth & Brooks/Cole, Pacific Grove. · Zbl 0699.62001
[8] Castaing, C., Le théorème de Dunford-Pettis généralisé, C. R. acad. sci. Paris, Sér. A, 268, 327-329, (1969) · Zbl 0184.40203
[9] Cox, E., 1994. The Fuzzy Systems Handbook. Academic Press, Cambridge. · Zbl 0847.68111
[10] Debreu, G., 1966. Integration of correspondences. Proc. 5th Berkeley Symp. on Math. Statist. and Probability, vol. II, Part I, pp. 351-372.
[11] Dudewicz, E.J., Mishra, S.N., 1988. Modern Mathematical Statistics. Wiley, New York.
[12] Gil, M.A.; López-Dı́az, M., Fundamentals and Bayesian analyses of decision problems with fuzzy-valued utilities, Int. J. approx. reason., 15, 203-224, (1996) · Zbl 0949.91504
[13] Gil, M.A., López-Dı́az, M., Rodrı́guez-Muñiz, L.J., 1998. An improvement of a comparison of experiments in statistical decision problems with fuzzy utities. IEEE Trans. Systems. Man, Cybernet., in press.
[14] Hiai, F.; Umegaki, H., Integrals, conditional expectations, and martingales of multivalued functions, J. multivariate anal., 7, 149-182, (1977) · Zbl 0368.60006
[15] Himmelberg, C.J., Measurable relations, Fund. math., 87, 53-72, (1975) · Zbl 0296.28003
[16] López-Dı́az, M.; Gil, M.A., Constructive definitions of fuzzy random variables, Statist. probab. lett., 36, 135-143, (1997) · Zbl 0929.60005
[17] López-Dı́az, M., Gil, M.A., 1998. The \(λ\)-average value of the expected value of a fuzzy random variables. Fuzzy Sets and Systems, in press.
[18] Lubiano, M.A., Gil, M.A., López-Dı́az, M., López, M.T., 1998. The \(λ\)-mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets and Systems, in press.
[19] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[20] Stojaković, M., Fuzzy conditional expectation, Fuzzy sets and systems, 52, 53-60, (1992) · Zbl 0782.60009
[21] Thompson, S.K., 1992. Sampling. Wiley, New York.
[22] Zadeh, L.A., 1975. The concept of a linguistic variable and its application to approximate reasoning. Inform. Sci. Part 1 8, 199-249; Part 2 8, 301-353; Part 3 9, 43-80. · Zbl 0397.68071
[23] Zadeh, L.A., A fuzzy-algorithmic approach to the definition of complex or imprecise concepts, Int. J. man-Mach. stud., 8, 249-291, (1976) · Zbl 0332.68068
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.