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486 Introduction to Robust Estimation and Hypothesis Testing b 1 , . . . , b p so that | j | is approximately equal to zero. Note that this approach can be used ^ to generalize the TheilSen estimator by replacing Kendall's tau with any reasonable correlation coefficient. The relative merits of this third way of extending the TheilSen estimator to multiple predictors have not been explored. Results regarding the small-sample efficiency of the GaussSeidel method, versus using randomly sampled elemental subsets, are reported in Wilcox (2004d). The choice of method can make a practical difference, but currently there is no compelling reason to prefer one method over the other based solely on efficiency. A criticism of method TSG is that as p increases, its finite sample breakdown point decreases (Rousseeuw & Leroy, 1987, p. 148). Another possible concern is that the marginal medians are location equivariant but not affine equivariant. (See Eq. (6.9) for a definition of affine equivariance when referring to a multivariate location estimator.) A regression estimator T is affine equivariant if for any nonsingular matrix A, T (x i A, y i ; i = 1, . . . , n) = A -1 T(xi , y i ; i = 1, . . . , n). Because the marginal medians are not affine equivariant, TSG is not affine equivariant either. Yet one more criticism is that with only (n 2 - n)/2 randomly sampled elemental subsets, if n is small, rather unstable results can be obtained, meaning that if a different set of (n 2 - n)/2 elemental subsets is used, the estimates can change substantially. If, for example, n = 20 and p = 2, only 190 resamples are used from among the 1140 elemental subsets. (Of course, when n is small, it is a simple matter to increase the number of sampled elemental subsets, but just how many additional samples should be taken has not been investigated.) A regression estimator T is regression equivariant if for any vector v, T (x i , y i + xi v; i = 1, . . . , n) = T(xi , y i ; i = 1, . . . , n) + v. And T is said to be scale equivariant if T (x i , cy i ; i = 1, . . . , n) = cT (x i , y i ; i = 1, . . . , n). It is noted that method TS also fails to achieve affine equivariance, but it does achieve regression equivariance and scale equivariance. 10.2.1 R Functions tsreg, correg, and regplot The R function tsreg(x,y,iter=10) www.elsevierdirect.com