510 Introduction to Robust Estimation and Hypothesis Testing The R function COVreg(x,y,cov.fun=MARest,loc.fun=MARest,xout=F,outfun=out,...) estimates the slopes and intercept via Eqs (10.11) and (10.12) without iterations and defaults to using Marrona's M-estimator in Section 6.3.13. I Example For the star data shown in Figure 6.3 of Chapter 6, bireg estimates the slope and intercept to be 2.66 and -6.7, respectively. The OLS estimates are -0.41 and 6.79. The function winreg estimates the slope to be 0.31 using 10% Winsorization (tr=0.1), and this is considerably smaller than the estimate of 2.66 returned by bireg. Also, winreg reports a warning message that convergence was not obtained in 20 iterations. This problem seems to be very rare. Increasing iter to 50, convergence is obtained, and again the slope is estimated to be 0.31, but the estimate of the intercept drops from 3.62 to 3.61. Using the default 20% Winsorization (tr=0.2), the slope is now estimated to be 2.1 and convergence problems are eliminated. I 10.13.7 L-Estimators A reasonable approach to regression is to compute some initial estimate of the parameters, compute the residuals, and then re-estimate the parameters based in part on the trimmed residuals. This strategy was employed by Welsh (1987a,b) and expanded upon by De Jongh, De Wet, and Welsh (1988). The small-sample efficiency of this approach does not compare ^ well with other estimators such as m or M regression with Schweppe weights (Wilcox, 1996d). In terms of achieving high efficiency when there is heteroscedasticity, comparisons with the better estimators in this chapter have not been made. So, even though the details of the method are not described here, it is not being suggested that Welsh's estimator be abandoned. 10.13.8 L 1 and Quantile Regression Yet another approach is to estimate the regression parameters by minimizing |r i |, the so-called L 1 norm, which is just the sum of the absolute values of the residuals. This approach predates OLS by 50 years. This is, of course, a special case of the LTA estimator in Section 10.5. The potential advantage of L 1 (or least absolute value) regression over OLS, in terms of efficiency, was known by Laplace (1818). The L 1 approach reduces the influence of outliers, but the breakdown point is still 1/n. More precisely, L 1 regression protects against unusual y values, but not leverage points, which can have a large influence on the fit to data. www.elsevierdirect.com