Redescending M-estimator

In statistics, redescending M-estimators are Ψ-type M-estimators which have ψ functions that are non-decreasing near the origin, but decreasing toward 0 far from the origin. Their ψ functions can be chosen to redescend smoothly to zero, so that they usually satisfy ψ(x) = 0 for all x with |x| > r, where r is referred to as the minimum rejection point. Due to these properties of the ψ function, these kinds of estimators are very efficient, have a high breakdown point and, unlike other outlier rejection techniques, they do not suffer from a masking effect. They are efficient because they completely reject gross outliers, and do not completely ignore moderately large outliers (like median). Redescending M-estimators have high breakdown points (close to 0.5), and their Ψ function can be chosen to redescend smoothly to 0. This means that moderately large outliers are not ignored completely, and greatly improves the efficiency of the redescending M-estimator. The redescending M-estimators are slightly more efficient than the Huber estimator for several symmetric, wider tailed distributions, but about 20% more efficient than the Huber estimator for the Cauchy distribution. This is because they completely reject gross outliers, while the Huber estimator effectively treats these the same as moderate outliers. As other M-estimators, but unlike other outlier rejection techniques, they do not suffer from masking effects. The M-estimating equation for a redescending estimator may not have a unique solution. Consequently, the initial point for an iterative solution must be chosen with care, e.g., by use of another estimator. When choosing a redescending Ψ function, care must be taken such that it does not descend too steeply, which may have a very bad influence on the denominator in the expression for the asymptotic variance where F is the mixture model distribution. This effect is particularly harmful when a large negative value of ψ′(x) combines with a large positive value of ψ2(x), and there is a cluster of outliers near x.

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Redescending M-estimators

Stephan Morgenthaler

in finite sample studies redescending M-estimators outperform bounded M-estimators (see for example, Andrews et al. [1972. Robust Estimates of Location. Princeton University Press, Princeton]). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we study the optimality of redescending M-estimators. We show that redescending M-estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing over a class of densities the minimum variance sensitivity over a class of estimators. As a particular result, we prove that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides a guaranteed level of an estimator's variance sensitivity over the class of densities with a bounded variance. (C) 2007 Elsevier B.V. All rights reserved.

Elsevier2008