Scale Equivariant Alternative for Fuzzy M-Estimators of Location

Abstract

The Aumann-type mean fulfills very convenient properties as a location measure of a random fuzzy number, but its high sensitivity to outliers makes other alternatives, such as fuzzy M-estimators of location, more suitable to describe contaminated data sets. Under some conditions, fuzzy M-estimators fulfill properties such as the strong consistency and the translation equivariance. However, the scale equivariance does not hold in general and the choice of the measurement units may have too much influence on the results. A first solution to solve this was the selection of the tuning parameters involved in the most used loss functions (Huber’s, Tukey’s and Hampel’s) in terms of the distribution of distances of the observed data to the considered initial location estimate. Now a second solution is proposed including a robust estimate of the unknown dispersion in the definition of fuzzy M-estimators of location. The empirical comparison of both proposals shows that the latter solution may be more suitable for dealing with extreme data, and therefore it could better identify which observations should be considered outliers indeed.

This paper is dedicated to the memory of Prof. Pedro Gil, who not only taught my mother and I Statistics in an interesting and calm way, but also left me bright memories in relation to our condition of neighbours and the conferences we both attended, as well as the Champanadas he cheered up with his accordion. I am deeply grateful for such moments and lessons.