M-estimators with asymmetric influence functions: the distribution case

Abstract

Many applications consecrate the use of asymmetric distributions, and practical situations often require robust parametric inference. This paper presents the derivation of M-estimators with asymmetric influence functions, motivated by the distribution. This law, regarded as the universal model for speckled imagery, can be highly skewed and maximum likelihood estimation can be severely hampered by small percentages of outliers. These outliers appear mainly because the hypothesis of independence and equal distribution of observations are seldom satisfied in practice; for instance, in the process of filtering, some pixels within a window frequently come from regions with different underlying distributions. Traditional robust estimation methods, on the basis of symmetric robustifying functions, assume that the distribution is symmetric, but when the data distribution is asymmetric, these methods yield biased estimators. Empirical influence functions for maximum likelihood estimators are computed, and based on this information we propose the asymmetric M-estimator (AM-estimator), an M-estimator with asymmetric redescending functions. The performance of AM estimators is assessed, and it is shown that they either compete with or outperform both maximum likelihood and Huber-type M-estimators.