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10.2.1 R Functions tsreg, correg, and re... > 10.2.1 R Functions tsreg, correg, an... - Pg. 486

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