Tukey’s biweight loss function for fuzzy set-valued M-estimators of location

Abstract

The Aumann-type mean is probably the best-known measure for the location of a random fuzzy set. Despite its numerous probabilistic and statistical properties, it inherits from the mean of a real-valued random variable the high sensitivity to outliers or data changes. Several alternatives extending the concept of median to the fuzzy setting have already been proposed in the literature. Recently, the adaptation of locationM-estimators has also been tackled. The expression of fuzzy-valued location M-estimators as weighted means under mild conditions allows us to guarantee that these measures take values in the space of fuzzy sets. It has already been shown that these conditions hold for the Huber and Hampel families of loss functions. In this paper, the strong consistency and the maximum finite sample breakdown point when the Tukey biweight (or bisquare) loss function is chosen are analyzed. Finally, a real-life example will illustrate the influence of the choice of the loss function on the outputs.