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6.3 Some Affine Equivariant Estimators > 6.3.9 Median Ball Algorithm - Pg. 231

Chapter 6 Some Multivariate Methods 231 For general theoretical results on this approach to estimation, see Tyler (1994). Properties of certain variations were reported by Maronna and Yohai (1995). Also see Arcones, Chen, & Gine (1994), Bai and He (1999), He and Wang (1997), Donoho and Gasko (1992), Gather and Hilker (1997), Zuo (2003), Zuo, Cui, and He (2004), and Zuo, Cui, and Young (2004). Gervini (2002) derived the influence function assuming that sampling is from an elliptical distribution. (For an extension of M-estimators to the multivariate case that has a high breakdown point and deals with missing values, see Chen & Victoria-Feser, 2002.) Zuo, Hengjian, and He (2004) and Zuo, Hengjian, and Young (2004) suggest a particular variation of the Donoho­Gasko W-estimator for general use. Let P i be the projection depth of x i described at the end of Section 6.2.7. Let C be the median of the P i values. If P i < C, set w i = exp[-K (1 - P i /C) 2 ] - exp(-K ) , 1 - exp(-K ) otherwise w i = 1, and the measures of location and scatter are given by Eqs. (6.13) and (6.14), respectively. From Zuo et al., setting the constant K = 3 results in good asymptotic efficiency, relative to the sample mean, under normality.