Location-Free Robust Scale Estimates for Fuzzy Data

In analyzing fuzzy-valued imprecise data statistically, scale measures/estimates play an important role. Scale measures/estimates of data sets are often considered, among others, to descriptively summarize them, to compare the dispersion or the spread of different data sets, standardize data, state rules for detecting outliers, formulate regression objective functions, etc. To be robust, an estimate of scale should have a finite breakdown point close to 50% (i.e., around half data should be replaced by "outliers" to make the estimate break down, either in the sense of exploding to infinity or imploding to zero). In this respect, the median distance deviation about the median (MDD) for fuzzy data sets has already been introduced and its robust behavior has been proved. In contrast to the real-valued case, computation of the MDD for fuzzy data is much more complex and cannot be exactly but approximately performed in general. These computational inconveniences are mainly associated with the fact that, in general, the "median of the fuzzy data set" cannot be exactly calculated, but simply approximated through some levels, and it does not preserve the shape of the fuzzy data. The same happens with the distances between data and the approximate median. Consequently, the use of location-free scale measures would be especially appropriate-to-use in this fuzzy-valued environment. This article aims to extend some robust global scale estimates, and to prove that the extension remains robust. Furthermore, it will be shown that these estimates can be easily and exactly computed for fuzzy trapezoidal data, the assumption of considering trapezoidal data not implying an important loss of generality in the setting of scale estimation.