234 Introduction to Robust Estimation and Hypothesis Testing 6.3.13 An M-Estimator As noted at the beginning of this section, a concern about (affine equivariant) M-estimators is that they have a breakdown point of at most 1/( p + 1). Also, Devlin, Gnanadesikan, and Kettenring (1981, p. 361) found that M-estimators could tolerate even fewer outliers than indicated by this upper bound. Despite this, in situations where p is small, this approach might be deemed satisfactory. For example, Zu and Yuan (2010) suggest an approach to a mediation analysis that is based in part on an M-estimator with Huber weights, which was derived by Maronna (1976). A slight modification of the Zu and Yuan method has been found to perform relatively well in simulations, so for completeness, Maronna's M-estimator is outlined here. (Details of Zu and Yuan method for performing a mediation analysis are outlined in Section 11.7.2.) The computation of this estimator is accomplished via an iterative scheme that corresponds to a multivariate version of the W-estimator in Section 3.8. Roughly, an initial estimate of the ¯ mean and covariance matrix is computed, which here is taken to be usual mean X vector and covariance matrix S. Based on this initial estimate, squared Mahalanobis distances are computed: ¯ ¯ d i 2 = (X i - X) S -1 (X i - X). Imagine that one wants to downweight a proportion of the observations. Let 2 be the 1 - quantile of a chi-squared distribution with p degrees of freedom. Let w i = 1 if d i ; otherwise w i = /d i . Then an updated estimate of the mean and covariance matrix is given by ¯ X = and S = 1 n ¯ ¯ w i 2 (X i - X)(X i - X) , w i X i /n respectively, where is chosen so that S is an unbiased estimate of the covariance matrix under normality. These updated estimates are used to update the squared Mahalanobis distances, which in turn yields a new updated estimate of the mean and covariance matrix. This process is continued until convergence is achieved. 6.3.14 R Function MARest The R function MARest(x,kappa=0.1) www.elsevierdirect.com