Empirical Comparison of the Performance of Location Estimates of Fuzzy Number-Valued Data

Abstract

Several location measures have already been proposed in the literature in order to summarize the central tendency of a random fuzzy number in a robust way. Among them, fuzzy trimmed means and fuzzy M-estimators of location extend two successful approaches from the realvalued settings. The aim of this work is to present an empirical comparison of dierent location estimators, including both fuzzy trimmed means and fuzzy M-estimators, to study their dierences in nite sample behaviour.

Empirical analysis of the maximum asymptotic bias of location estimators for fuzzy number-valued data

“…By empirically comparing maxbias curves and asymptotic biases, we have come to the conclusion that the behaviour of the fuzzy trimmed mean with β = 0.5 and the Hampel fuzzy M-estimator of location should be highlighted among the fuzzy-valued location measures. Indeed, the performance of the fuzzy trimmed means with a high trimming level and the Hampel fuzzy M-estimator of location was already commended before because they yielded the best solutions in many situations considered in the previously published comparative analysis of the finite sample behaviour of fuzzy number-valued location measures [9]. The fuzzy trimmed means with β = 0.5 often yields the lowest asymptotic bias in the developed simulation studies (e.g., when C D ≥ 2), but their high asymptotic variance, in contrast to the minimum asymptotic variance achieved by fuzzy M-estimators of location, is a high price to pay.…”

“…For the empirical study in Section 4, we have chosen the Huber and Hampel families of loss functions in order to complement the simulations presented in [9]. The Huber loss function is given by…”

“…Under general conditions (see [4]), Huber and Hampel fuzzy M-estimators also obtain the maximum finite sample breakdown point, whereas fuzzy trimmed means will only reach the maximal value 1 n ⌊ n+1 2 ⌋ for trimming proportion β = 0.5. In [9], the bias, the variance and the mean squared error of all these location measures (also including the Aumann-type mean) were computed with the aim of comparing the finite-sample behaviour of the estimators. No uniformly most appropriate location estimate could be chosen, as in the classical settings, but it could be concluded that Huber and Hampel fuzzy M-estimators in general behave well for small contamination levels while fuzzy trimmed means performed better for larger contamination proportions.…”

“…[3,4,5,6,7,8]). Fuzzy trimmed means, Huber and Hampel fuzzy M-estimates, the 1-norm median and the wabl/ldev/rdev median have been empirically compared by means of their bias, variance and mean squared error in [9]. As in the classical settings, it has been shown that there is no uniformly optimal location estimator.…”