Advantages of M-estimators of location for fuzzy numbers based on Tukey's biweight loss function

Abstract

M-estimators of location have been adapted to summarize the central tendency of random fuzzy numbers in a robust way. Under mild conditions on the loss function, which include the well-known Huber and Hampel families of loss functions, fuzzy number-valued location M-estimators exist and can be expressed as weighted means of the observations. Huber and Hampel loss functions depend on one and three tuning parameters, respectively. Some empirical analyses have been developed to compare the finite-sample behavior of the corresponding location M-estimators when these tuning parameters are well-chosen quantiles of the distribution of distances from an initial estimate to each observation. In that sense, it has been shown that the flexibility of the three tuning parameters in the Hampel loss function makes the corresponding M-estimator more accurate than the M-estimator based on the Huber loss function in many situations. More recently, Tukey's biweight (or bisquare) loss function has also been used to compute M-estimators of location for random fuzzy numbers. The robustness of all these estimators has been proven by their finite sample breakdown point, but a simulation study to compare the finite-sample behavior of the M-estimators defined in terms of Hampel and Tukey loss functions is still lacking. This paper aims to develop such simulation results and analyze the advantages of choosing the Tukey loss function.