Table 2. Empirical powers for six estimation procedures in different combinations of simulation parameters.
True cost for one unit of effectiveness = 10 | ||||||||||||||||||||||||||
Outlier Distribution | I. N(150,1) | II. N(200,1) | III. N(150,1) or N(200,1) | |||||||||||||||||||||||
Outlier Proportion | 5% | 10% | 20% | 30% | 5% | 10% | 20% | 30% | 5% | 10% | 20% | 30% | ||||||||||||||
WTP | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | 12 | 13 | ||
Sample size | 100 | OLS estimation | 0.12 | 0.17 | 0.13 | 0.14 | 0.13 | 0.17 | 0.13 | 0.14 | 0.09 | 0.11 | 0.09 | 0.11 | 0.10 | 0.12 | 0.12 | 0.12 | 0.11 | 0.13 | 0.10 | 0.13 | 0.11 | 0.13 | 0.11 | 0.12 |
Huber M-estimation | 0.21 | 0.32 | 0.20 | 0.29 | 0.15 | 0.20 | 0.11 | 0.12 | 0.22 | 0.30 | 0.22 | 0.30 | 0.13 | 0.20 | 0.10 | 0.11 | 0.22 | 0.33 | 0.18 | 0.29 | 0.13 | 0.18 | 0.08 | 0.09 | ||
Hampel M-estimation | 0.22 | 0.34 | 0.21 | 0.28 | 0.13 | 0.15 | 0.13 | 0.14 | 0.24 | 0.37 | 0.25 | 0.38 | 0.23 | 0.33 | 0.12 | 0.12 | 0.24 | 0.37 | 0.22 | 0.33 | 0.17 | 0.22 | 0.10 | 0.12 | ||
Tukey's bisquare M-estimation | 0.22 | 0.34 | 0.24 | 0.34 | 0.21 | 0.31 | 0.11 | 0.12 | 0.23 | 0.34 | 0.27 | 0.37 | 0.21 | 0.35 | 0.11 | 0.11 | 0.26 | 0.39 | 0.23 | 0.35 | 0.23 | 0.32 | 0.14 | 0.17 | ||
MM-estimation | 0.22 | 0.34 | 0.23 | 0.35 | 0.20 | 0.30 | 0.15 | 0.16 | 0.22 | 0.34 | 0.26 | 0.37 | 0.21 | 0.35 | 0.21 | 0.30 | 0.26 | 0.39 | 0.23 | 0.35 | 0.22 | 0.32 | 0.14 | 0.18 | ||
LTS estimation | 0.27 | 0.39 | 0.27 | 0.40 | 0.22 | 0.32 | 0.23 | 0.31 | 0.28 | 0.40 | 0.28 | 0.41 | 0.23 | 0.34 | 0.20 | 0.30 | 0.31 | 0.43 | 0.24 | 0.37 | 0.24 | 0.35 | 0.19 | 0.26 | ||
500 | OLS estimation | 0.34 | 0.49 | 0.25 | 0.38 | 0.32 | 0.43 | 0.36 | 0.43 | 0.22 | 0.36 | 0.21 | 0.29 | 0.18 | 0.25 | 0.22 | 0.26 | 0.27 | 0.41 | 0.22 | 0.31 | 0.22 | 0.28 | 0.25 | 0.32 | |
Huber M-estimation | 0.65 | 0.86 | 0.51 | 0.78 | 0.36 | 0.55 | 0.19 | 0.27 | 0.65 | 0.86 | 0.56 | 0.80 | 0.35 | 0.54 | 0.08 | 0.12 | 0.63 | 0.87 | 0.52 | 0.79 | 0.35 | 0.53 | 0.15 | 0.21 | ||
Hampel M-estimation | 0.69 | 0.90 | 0.53 | 0.77 | 0.15 | 0.25 | 0.36 | 0.43 | 0.71 | 0.91 | 0.70 | 0.90 | 0.61 | 0.87 | 0.22 | 0.26 | 0.71 | 0.90 | 0.62 | 0.83 | 0.38 | 0.59 | 0.24 | 0.30 | ||
Tukey's bisquare M-estimation | 0.72 | 0.91 | 0.66 | 0.91 | 0.60 | 0.83 | 0.20 | 0.29 | 0.70 | 0.90 | 0.70 | 0.90 | 0.60 | 0.87 | 0.12 | 0.16 | 0.70 | 0.91 | 0.66 | 0.89 | 0.67 | 0.88 | 0.49 | 0.50 | ||
MM-estimation | 0.72 | 0.91 | 0.66 | 0.91 | 0.61 | 0.83 | 0.49 | 0.51 | 0.70 | 0.90 | 0.70 | 0.90 | 0.60 | 0.88 | 0.58 | 0.82 | 0.71 | 0.91 | 0.67 | 0.89 | 0.67 | 0.88 | 0.53 | 0.67 | ||
LTS estimation | 0.73 | 0.90 | 0.67 | 0.91 | 0.60 | 0.86 | 0.62 | 0.83 | 0.72 | 0.91 | 0.73 | 0.89 | 0.61 | 0.87 | 0.59 | 0.83 | 0.71 | 0.91 | 0.68 | 0.90 | 0.69 | 0.90 | 0.60 | 0.82 | ||
1000 | OLS estimation | 0.51 | 0.78 | 0.50 | 0.70 | 0.48 | 0.63 | 0.58 | 0.71 | 0.34 | 0.49 | 0.27 | 0.41 | 0.30 | 0.37 | 0.32 | 0.40 | 0.40 | 0.64 | 0.39 | 0.53 | 0.33 | 0.46 | 0.44 | 0.54 | |
Huber M-estimation | 0.88 | 0.99 | 0.82 | 0.96 | 0.56 | 0.76 | 0.27 | 0.39 | 0.89 | 0.99 | 0.81 | 0.98 | 0.56 | 0.81 | 0.10 | 0.15 | 0.87 | 0.99 | 0.82 | 0.97 | 0.57 | 0.78 | 0.20 | 0.32 | ||
Hampel M-estimation | 0.92 | 0.99 | 0.82 | 0.96 | 0.24 | 0.40 | 0.58 | 0.71 | 0.93 | 1.00 | 0.91 | 0.99 | 0.90 | 0.99 | 0.32 | 0.40 | 0.91 | 1.00 | 0.91 | 0.99 | 0.56 | 0.83 | 0.40 | 0.51 | ||
Tukey's bisquare M-estimation | 0.93 | 1.00 | 0.92 | 0.99 | 0.87 | 0.99 | 0.32 | 0.43 | 0.92 | 1.00 | 0.90 | 0.99 | 0.89 | 0.99 | 0.14 | 0.21 | 0.92 | 1.00 | 0.93 | 1.00 | 0.86 | 0.99 | 0.77 | 0.69 | ||
MM-estimation | 0.93 | 1.00 | 0.92 | 0.99 | 0.87 | 0.99 | 0.71 | 0.75 | 0.92 | 1.00 | 0.90 | 0.99 | 0.89 | 0.99 | 0.86 | 0.97 | 0.92 | 1.00 | 0.93 | 1.00 | 0.86 | 0.99 | 0.79 | 0.91 | ||
LTS estimation | 0.92 | 0.99 | 0.92 | 0.99 | 0.87 | 0.99 | 0.86 | 0.98 | 0.93 | 1.00 | 0.90 | 0.99 | 0.90 | 0.99 | 0.87 | 0.93 | 0.92 | 1.00 | 0.93 | 1.00 | 0.86 | 0.99 | 0.86 | 0.98 |