M-Estimates of Location for the Robust Central Tendency of Fuzzy Data

Abstract

The Aumann-type mean has been shown to possess valuable properties as a measure of the location or central tendency of fuzzy data associated with a random experiment. However, concerning robustness its behaviour is not appropriate. The Aumann-type mean is highly affected by slight changes in the fuzzy data or when outliers arise in the sample. Robust estimators of location, on the other hand, avoid such adverse effects. For this purpose, this paper considers the M-estimation approach and discusses conditions under which this alternative yields valid fuzzy-valued M-estimators. The resulting M-estimators are applied to a real-life example. Finally, some simulation studies show empirically the suitability of the introduced estimators. Index Terms-fuzzy number-valued data, M-estimators, random fuzzy numbers, robust location of fuzzy data.

Scale Equivariant Alternative for Fuzzy M-Estimators of Location

“…Among the robust location measures for fuzzy-valued data, the performance of fuzzy M-estimators of location is certainly remarkable, achieving the best results in many of the situations studied in [10]. Definition 6.…”

“…This is not the only robust location measure for fuzzy numbers proposed in the literature (we could think, for example, about some extensions of the concept of median like the ones introduced in [11] and [12]), but their performance seems to be the best in general. Although the empirical study addressed in [10] concludes that there is no uniformly best location estimator, it highlights the good behavior of fuzzy M-estimators of location.…”

“…Even when the Aumann-type mean fulfills very good statistical and probabilistic properties, outliers have too much impact on its estimate. For that reason, M-estimators of location have been recently defined in the fuzzy number-valued case by Sinova et al [10] and their robustness has been shown. This is not the only robust location measure for fuzzy numbers proposed in the literature (we could think, for example, about some extensions of the concept of median like the ones introduced in [11] and [12]), but their performance seems to be the best in general.…”

“…This is an important drawback, since it means that measurement units could have a lot of impact on the results. A first solution was provided in [10] and consists in choosing the tuning parameters in the used loss functions (Huber's and Hampel's, and it could be also applied to other functions such as Tukey's) taking into account the distribution of distances from the observed data to the initial estimate considered for the computation of the corresponding M-estimator. Therefore, the loss function is adapted to the magnitude of the data we are working with.…”

Fuzzy data analysis and classification

“…Körner (2000), Montenegro et al (2001), Gil et al (2006), González-Rodríguez et al (2012), Ramos-Guajardo et al (2010), Ramos-Guajardo and Lubiano (2012) and Lubiano et al (2016b) Fuzzy estimates of location of random fuzzy numbers; robustness Lubiano and Gil (1999) and Sinova et al (2016) Statistical comparison of fuzzy scale with other imprecise-valued scales De la Rosa de Sáa et al (2016), Gil et al (2015) and Lubiano et al (2016aLubiano et al ( , 2017 Fuzzy inequality Gil et al (1998) Discriminant analysis Colubi et al (2011) Cluster analysis Hathaway et al (1996), Pedrycz et al (1998), Auephanwiriyakul and Keller (2002), D'Urso (2007) and Coppi et al (2012) Regression analysis Celminš (1987), Diamond (1988), Näther and Albrecht (1990) …”