Robust scale estimators for fuzzy data

Abstract

Observations distant from the majority or deviating from the general pattern often appear in datasets. Classical estimates such as the sample mean or the sample variance can be substantially affected by these observations (outliers). Even a single outlier can have huge distorting influence. However, when one deals with real-valued data there exist robust measures/estimates of location and scale (dispersion) which reduce the influence of these atypical values and provide approximately the same results as the classical estimates applied to the typical data without outliers. In reallife, data to be analyzed and interpreted are not always precisely defined and they cannot be properly expressed by using a numerical scale of measurement. Frequently, some of these imprecise data could be suitably described and modelled by considering a fuzzy rating scale of measurement. In this paper, several well-known scale (dispersion) Authors are very grateful for the insight comments from the reviewers of the original. 123 S. de la Rosa de Sáa et al. estimators in the real-valued case are extended for random fuzzy numbers (i.e., random mechanisms generating fuzzy-valued data), and some of their properties as estimators for dispersion are examined. Furthermore, their robust behaviour is analyzed using two powerful tools, namely, the finite sample breakdown point and the sensitivity curves. Simulations, including empirical bias curves, are performed to complete the study.

A hypothesis testing-based discussion on the sensitivity of means of fuzzy data with respect to data shape

“…A closely related open problem is that of analyzing the influence of the choice of the shape of fuzzy data associated with random experiments on other location/central tendency or dispersion/scale measures of the distribution of random fuzzy numbers, like for instance, the robust measures by Sinova et al [39,40,41] and De la Rosa de Sáa et al [14].…”

Scale Equivariant Alternative for Fuzzy M-Estimators of Location

“…A second alternative is introduced in Section 3 of this paper by extending the classical M-estimators of location with unknown dispersion, which are based on a robust estimate of the dispersion. The median distance deviation about the median of a random fuzzy number has been defined using the ρ 1 distance, which is an L 1 metric based on the infimum/supremum characterization of fuzzy numbers, in [4]. However, a new alternative is now considered, by replacing the ρ 1 distance by the D ϕ θ metric, since fuzzy M-estimators of location are defined in terms of the latter (due to the isometrical embedding mentioned above).…”

“…In the classical settings, where the same problem has had to be dealt with, a robust estimate of the dispersion is introduced in the definition of the M-estimator of location to make it scale equivariant. Recently, a robust estimate of the dispersion of a random fuzzy number, the median distance deviation about the median, has been analyzed (see [4]). The idea is, in consequence, to use a similar median distance deviation about the median to extend M-estimators of location with unknown dispersion to the fuzzy number-valued settings.…”

Case Study-Based Sensitivity Analysis of Scale Estimates w.r.t. the Shape of Fuzzy Data

“…The already developed discussions concern location of the random processes generating fuzzy data (see Lubiano et al [7,9]), and a few ones regard the Fréchet-type variance of these processes (see De la Rosa de Sáa et al [2,3]).…”

“…This paper presents a discussion involving some scale estimates for fuzzy data sets that have been recently introduced (see [3]). The discussion is to be based on a case study and will include both, descriptive and inferential conclusions.…”