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Regression and correlation analyses of a linear relation between random intervals. (English) Zbl 0981.62062

Summary: We develop regression and correlation analyses of a certain general linear relation between two random elements whose values are non-empty compact intervals. To this purpose, we firstly extend the least-squares method to deal with the involved random elements on the basis of a generalized metric defined on the space of the considered intervals. As a complementary study, a coefficient quantifying the strength of the linear relation between the two random elements is also presented, and a discussion of the extreme values for this measure is presented. A real-life example illustrates these studies.

MSC:

62J05 Linear regression; mixed models
62J99 Linear inference, regression
60D05 Geometric probability and stochastic geometry
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[1] Aumann, R.J. (1965). Integrals of set-valued functions.Journal of Mathematical Analysis and Applications,12, 1–12. · Zbl 0163.06301
[2] Bandemer, H. and S. Gottwald (1995).Fuzzy Sets, Fuzzy Logic, Fuzzy Methods with Applications. John Wiley, Chichester. · Zbl 0833.94028
[3] Bandemer, H. and W. Näther (1992).Fuzzy Data Analysis. Kluwer Academic Publications, Dortrecht. · Zbl 0776.94021
[4] Bertoluzza, C., N. Corral and A. Salas (1995) On a new class of distances between fuzzy numbers. Mathware & Soft Computing.2, 71–84. · Zbl 0887.04003
[5] Cressic, N.A.C. (1993).Statisticas for Spatial Data. John Wiley, New York.
[6] Debreu, A. (1967). Integration of correspondences.Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics and Probability, 351–372. · Zbl 0211.52803
[7] Diamond, P. (1988). Fuzzy least squares.Inform. Sci. 46, 141–157. · Zbl 0663.65150
[8] Diamond, P. (1990). Least squares fitting of compact set-valued data.Journal of Mathematical Analysis and Applications,147, 531–544. · Zbl 0704.65006
[9] Diamond, P. and P.E. Kloeden (1994).Metric Space of Fuzzy Sets. World Sciences, New Jersey. · Zbl 0873.54019
[10] Hiai, F., and H. Umegaki (1977). Integrals, conditional expectations and martingales of multivalued functionsJournal of Multivariate Analysis,7, 149–182. · Zbl 0368.60006
[11] Himmelberg, C. J. (1975). Measurable relations.Fund. Math. 87, 53–72. · Zbl 0296.28003
[12] Kendall, D.G. (1974). Foundations of a theory of random sets. InStochastic Geometry, 3–9 (E.F. Harding and D.G. Kendall, eds.) John Wiley, New York. · Zbl 0275.60068
[13] Klement, E.P., M.L. Puri and D.A. Ralescu (1986). Limit theorems for fuzzy random variables.Proceedings of the Royal Society, London A,407, 171–182. · Zbl 0605.60038
[14] Körner, R. (1997a). On the variance of fuzzy random variables.Fuzzy Sets and Systems,92, 83–93. · Zbl 0936.60017
[15] Körner, R. (1997b). Linear Models with Random Fuzzy Variables. PhD Thesis. Freiberg University of Mining and Technology.
[16] Körner, R. and W. Näther (1998). Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates. Inform. Sci.109, 95–118. · Zbl 0930.62072
[17] López-Díaz, M. and M. A. Gil (1997). Constructive definitions of fuzzy random variables.Statistics and Probability Letters,36, 135–143. · Zbl 0929.60005
[18] López-Díaz, M. and M.A. Gil (1998a) The {\(\lambda\)}-average value and the fuzzy expectation of a fuzzy random variable.Fuzzy Sets and Systems,99, 347–352. · Zbl 0941.60030
[19] López-Díaz, M. and M.A. Gil (1998b) Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications.Journal of Statistical Planning and Inference,74, 11–29. · Zbl 0962.62005
[20] Lyashenko, N.N. (1980). The statistics of random compacts in the Euclidean space.Zap. Auchn. Semin. Leningr. Otd. Mat. Inst. Steklova 98, 115–139. · Zbl 0457.62016
[21] Lubiano, M.A. (1999).Variation Measures for Imprecise Random Elements,PhD Thesis. University of Oviedo.
[22] Matheron, G. (1975).Random Sets and Integral Geometry. John Wiley, New York. · Zbl 0321.60009
[23] Molchanov, I.S. (1993).Limit Theorems for Unions of Random Closed Sets. Lecture Notes in Mathematics,1561. Springer-Verlag, Berlin. · Zbl 0790.60015
[24] Näther, W. (1997). Linear statistical inference for random fuzzy data.Statistics,29, 221–240. · Zbl 1030.62530
[25] Näther, W. and M. Albrecht (1990). Linear regression with random fuzzy observations.Statistics,21, 521–531. · Zbl 0714.62063
[26] Puri, M.L. and D.A. Ralescu (1985). The concept of normality for fuzzy random variables.Annals of Probability.13, 1373–1379. · Zbl 0583.60011
[27] Puri, M.L. and D. Ralescu (1986). Fuzzy random variables.Journal of Mathematical Analysis and Applications,114, 409–422. · Zbl 0592.60004
[28] Salas, A., C. Bertoluzza and N. Corral (1991). Fuzzy Linear Regression: Existence of Solution for a generalized least squares method.Proceddings of the Fourth IFSA Conference, CMS, 233–235.
[29] Stoyan, D. (1998). Random sets: Models and Statistics.International Statistical Review,66, 1–27. · Zbl 0906.60006
[30] Stoyan, D., W.S. Kendall and J. Mecke (1995).Stochastic Geometry and its Applications, second edition. John Wiley, Chichester. · Zbl 0838.60002
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