On Asymptotic Distributions of Estimators Concerned with Fuzzy Random Data as Vague Perceptions of Crisp Random

Abstract

In this paper, the author investigates the asymptotic distributions of estimators concerned with the class of fuzzy random sets, which is considered as a model of the capricious vague perceptions of a crisp random phenomenon. First, the class of fuzzy random sets, which has been proposed by author as a model of the capricious vague perception of a crisp random phenomenon, is refined from the practical point of view. Secondly, using the refined class of fuzzy random sets, the expectations of fuzzy random sets are also refined. Applying the standard strong law of large numbers for the random elements in a separable Banach space, the convergence properties of estimators for expectations of refined random fuzzy sets are examined. Finally, asymptotic distributions concerned with the estimators are also investigated, applying the central limit theorem for the random elements in a separable Banach space. The theoretical aspect of this research for investing asymptotic properties of proposed estimators is mainly inspired by the papers of Krätschmer[1, 2, 3, 4, 5].