Limit distributions of least squares estimators in linear regression models with vague concepts

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have lately been introduced (cf. [V. Kratschmer, Induktive Statistik auf Basis unscharfer Meszkonzepte am Beispiel linearer Regressionsmodelle, unpublished postdoctoral thesis, Faculty of Law and Economics of the University of Saarland, Saarbrucken, 2001; V. Kratschmer, Least squares estimation in linear regression models with vague concepts, Fuzzy Sets and Systems, accepted for publication]) to improve the empirical meaningfulness of the relationships between the involved items by a more sensitive attention to the problems of data measurement, in particular, the fundamental problem of adequacy. The parameters of such models are still real numbers, and a method of estimation can be applied which extends directly the ordinary least squares method. In another recent contribution (cf. [V. Kratschmer, Strong consistency of least squares estimation in linear regression models with vague concepts, J. Multivar. Anal., accepted for publication]) strong consistency and n-consistency of this generalized least squares estimation have been shown. The aim of the paper is to complete these results by an investigation of the limit distributions of the estimators. It turns out that the classical results can be transferred, in some cases even asymptotic normality holds.