A linear regression model for imprecise response

Authors:
M. B. Ferraro;R. Coppi;G. González Rodríguez;A. Colubi
Affiliations:
Dipartimento di Statistica, Probabilità e Statistiche Applicate, Sapienza Università di Roma, Italy;Dipartimento di Statistica, Probabilità e Statistiche Applicate, Sapienza Università di Roma, Italy;European Centre for Soft Computing, Mieres, Spain;Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, Spain
Venue:
International Journal of Approximate Reasoning
Year:
2010

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Abstract

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R^3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.