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M-estimators and trimmed means: from Hilbert-valued to fuzzy set-valued data. (English) Zbl 07363874

Summary: Different approaches to robustly measure the location of data associated with a random experiment have been proposed in the literature, with the aim of avoiding the high sensitivity to outliers or data changes typical for the mean. In particular, M-estimators and trimmed means have been studied in general spaces, and can be used to handle Hilbert-valued data. Both alternatives are of interest due to their success in the classical framework. Since fuzzy set-valued data can be identified with a convex cone of a separable Hilbert space, the previous concepts have been recently applied to the one-dimensional fuzzy case. The aim of this paper is to extend M-estimators and trimmed means to \(p\)-dimensional fuzzy set-valued data, and to theoretically prove that they inherit robustness from the real settings. Some of such theoretical results are more general and directly apply to Hilbert-valued estimators and, in consequence, to functional data. A real-life example will also be included to illustrate the computation and behaviour of these estimators under contamination.

MSC:

62G35 Nonparametric robustness
62-07 Data analysis (statistics) (MSC2010)
03E72 Theory of fuzzy sets, etc.
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