Strong consistency and robustness of fuzzy medoids

Abstract

Central tendency of fuzzy number-valued data can be robustly summarized with different proposals from the literature, namely, fuzzy-valued medians, trimmed means and M-estimators of location. In many applications, fuzzy numbers of a specific shape, such as trapezoidal or triangular, are considered, since the chosen shape scarcely affects the value of these summary measures, whenever the ‘meaning’ is basically preserved. Whereas, irrespective of the considered data shape, M-estimators of location under the conditions of the representer theorem and trimmed means would share the same shape, fuzzy medians do not have to. Fuzzy medians must be frequently approximated through the computation of some of their α-levels, whence methods based on them become more complex computationally. All this might discourage users from choosing these measures to describe central tendency. Fuzzy medoids have been recently introduced as an alternative that keeps both the shape of the data and the idea inspiring fuzzy medians, by focusing on the minimization of the mean distance to sample observations, but constrained to the set of fuzzy-valued data. Consequently, it is guaranteed that they always coincide with a sample observation, like it happens (or can be assumed, by convention, to happen) with the median in real-valued scenarios. This work shows the strong consistency of fuzzy medoids as estimators of the corresponding population median (with respect to the same distance) and their robustness in terms of the finite sample breakdown point. Furthermore, some simulation studies have been developed to compare the finite-sample behaviour of fuzzy medoids and other robust central tendency measures.