In quantifying the central tendency of the distribution of a random fuzzy number (or fuzzy random variable in Puri and Ralescu's sense), the most usual measure is the Aumann-type mean, which extends the mean of a real-valued random variable and preserves its main properties and behavior. Although such a behavior has very valuable and convenient implications, 'extreme' values or changes of data entail too much influence on the Aumann-type mean of a random fuzzy number. This strong influence motivates the search for a more robust central tendency measure. In this respect, this paper aims to explore the extension of the median to random fuzzy numbers. This extension is based on the 1-norm distance and its adequacy will be shown by analyzing its properties and comparing its robustness with that of the mean both theoretically and empirically.

M-estimators of location are widely used robust estimators of the center of univariate or multivariate real-valued data. This paper aims to study M-estimates of location in the framework of functional data analysis. To this end, recent developments for robust nonparametric density estimation by means of M-estimators are considered. These results can also be applied in the context of functional data analysis and allow to state conditions for the existence and uniqueness of location M-estimates in this setting. Properties of these functional M-estimators are investigated. In particular, their consistency is shown and robustness is studied by means of their breakdown point and their influence function. The finite-sample performance of the M-estimators is explored by simulation. The M-estimators are also empirically compared to trimmed means for functional data.

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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.