Chapter 10 Robust Regression 509 Table 10.2: Estimates of R Using Covariance Methods, n = 20. VP 1 2 3 b mid 0.94 1.80 18.25 x and normal = 0.1 = 0.2 0.92 1.69 13.82 0.81 2.17 10.26 x normal, heavy-tailed b mid = 0.1 = 0.2 1.10 2.28 13.79 2.55 4.49 9.62 2.64 6.46 9.64 x heavy-tailed, normal b mid = 0.1 = 0.2 0.61 24.41 13.20 0.78 9.08 2.83 0.57 19.58 3.57 To provide some indication of how the efficiency of the biweight midregression and Winsorized regression methods compare to OLS regression, Table 10.2 shows estimates of R, the standard error of OLS regression divided by the standard error of the competing method. The columns headed by = 0.1 are the values when 10% Winsorization is used, and = 0.2 is 20%. These estimates correspond to three types of error terms: (x) = 1, (x) = x 2 , and (x) = 1/|x|. For convenience, these three choices are labeled VP1, VP2, and VP3. In general, the biweight and Winsorized regression methods compare well to OLS, and in some cases they offer a substantial advantage. Note, however, that when x has a heavy-tailed distribution, and is normal, OLS offers better efficiency when the error term is ^ ^ homoscedastic. In some cases, mid has better efficiency versus ad , but in other situations the