Tukey’s Biweight Loss Function for Fuzzy Set-Valued M-estimators of Location
The Aumann-type mean is probably the best-known measure for the location of a random fuzzy set. Despite its numerous probabilistic and statistical properties, it inherits from the mean of a real-valued random variable the high sensitivity to outliers or data changes. Several alternatives extending the concept of median to the fuzzy setting have already been proposed in the literature. Recently, the adaptation of location M-estimators has also been tackled. The expression of fuzzy-valued location M-estimators as weighted means under mild conditions allows us to guarantee that these measures take values in the space of fuzzy sets. It has already been shown that these conditions hold for the Huber and Hampel families of loss functions. In this paper, the strong consistency and the maximum finite sample breakdown point when the Tukey biweight (or bisquare) loss function is chosen are analyzed. Finally, a real-life example will illustrate the influence of the choice of the loss function on the outputs.
KeywordsRandom fuzzy set Robustness Location M-estimator Bisquare loss function Biweight loss function
This research has been partially supported by the Spanish Ministry of Economy and Competitiveness through the Grant MTM2013-44212-P, the Principality of Asturias/FEDER Grant GRUPIN14-101, Grant C16/15/068 of International Funds KU Leuven and IAP research network grant nr. P7/06 of the Belgian government. Their support is gratefully acknowledged.
- 3.Donoho DL, Huber PJ (1983) The notion of breakdown point. In: Bickel PJ, Doksum K, Hodges JL Jr (eds) A Festschrift for Erich L. Lehmann, WadsworthGoogle Scholar
- 12.Sinova B, Gil MA, Van Aelst S. M-estimates of location for the robust central tendency of fuzzy data. IEEE Trans Fuzzy Syst. AcceptedGoogle Scholar