TABLE 1.
Simulation results for Model I with the constant propensity score.
| Homogeneous Error | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Normal |
Log-Normal |
Cauchy |
||||||||
| n | method | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 |
| 100 | LS | 1.32 (0.040) | 80.7 | 1.06 | 2.36 (0.081) | 75.7 | 1.57 | 58.4 | 3.75 | |
| p(0.5) | 1.44 (0.042) | 80.1 | 1.13 | 1.73 (0.051) | 78.0 | 1.31 | 2.69 (0.077) | 75.2 | 1.63 | |
| p(0.25) | 1.90 (0.057) | 78.3 | 1.34 | 1.63 (0.051) | 79.0 | 1.29 | 5.29 (0.168) | 70.4 | 2.25 | |
| Huber | 1.15 (0.034) | 81.9 | 0.93 | 1.45 (0.044) | 79.9 | 1.13 | 2.61 (0.072) | 74.9 | 1.66 | |
| 200 | LS | 0.68 (0.021) | 85.6 | 0.59 | 1.10(0.033) | 82.0 | 0.91 | 58.7 | 3.70 | |
| p(0.5) | 0.73 (0.021) | 85.3 | 0.62 | 0.78 (0.021) | 84.1 | 0.70 | 1.23 (0.037) | 81.3 | 0.99 | |
| p(0.25) | 0.92 (0.028) | 84.0 | 0.75 | 0.70 (0.023) | 86.0 | 0.59 | 2.48 (0.079) | 75.7 | 1.64 | |
| Huber | 0.58 (0.017) | 86.8 | 0.50 | 0.66 (0.018) | 85.5 | 0.58 | 1.24 (0.035) | 80.8 | 1.03 | |
| 400 | LS | 0.33 (0.009) | 90.3 | 0.26 | 0.56 (0.016) | 87.1 | 0.46 | 59.2 | 3.61 | |
| p(0.5) | 0.35 (0.010) | 90.0 | 0.29 | 0.37 (0.010) | 89.0 | 0.34 | 0.56 (0.016) | 87.1 | 0.48 | |
| p(0.25) | 0.43 (0.013) | 89.1 | 0.34 | 0.33 (0.010) | 90.7 | 0.25 | 1.16(0.037) | 82.9 | 0.86 | |
| Huber | 0.28 (0.008) | 91.1 | 0.22 | 0.31 (0.009) | 90.2 | 0.27 | 0.58 (0.017) | 86.7 | 0.49 | |
| 800 | LS | 0.17 (0.005) | 93.2 | 0.13 | 0.26 (0.008) | 90.9 | 0.23 | 59.4 | 3.59 | |
| p(0.5) | 0.17 (0.005) | 93.1 | 0.13 | 0.19 (0.005) | 92.1 | 0.17 | 0.29 (0.009) | 90.7 | 0.24 | |
| p(0.25) | 0.22 (0.007) | 92.4 | 0.16 | 0.18 (0.006) | 93.6 | 0.12 | 0.59 (0.019) | 87.3 | 0.48 | |
| Huber | 0.14 (0.004) | 93.8 | 0.11 | 0.16 (0.005) | 93.1 | 0.14 | 0.29 (0.008) | 90.5 | 0.25 | |
| Heterogeneous Error | ||||||||||
|
Normal |
Log-Normal |
Cauchy |
||||||||
| n | method | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 |
| 100 | LS | 3.24 (0.110) | 74.7 | 1.70 | 8.98 (0.561) | 68.6 | 2.44 | 56.2 | 4.05 | |
| p(0.5) | 1.70 (0.060) | 80.5 | 1.08 | 1.80 (0.064) | 80.1 | 1.08 | 3.45 (0.124) | 75.1 | 1.69 | |
| p(0.25) | 2.50 (0.085) | 77.4 | 1.42 | 2.51 (0.079) | 76.8 | 1.46 | 9.13(0.341) | 67.2 | 2.66 | |
| Huber | 1.70 (0.057) | 80.4 | 1.10 | 1.87 (0.063) | 79.2 | 1.16 | 4.27 (0.155) | 72.8 | 1.93 | |
| 200 | LS | 1.54 (0.050) | 80.6 | 1.06 | 4.71 (0.244) | 73.4 | 1.85 | 55.2 | 4.17 | |
| p(0.5) | 0.78 (0.028) | 86.7 | 0.53 | 0.90 (0.032) | 85.3 | 0.63 | 1.49 (0.052) | 81.9 | 0.95 | |
| p(0.25) | 1.16(0.039) | 83.5 | 0.81 | 1.23 (0.039) | 82.0 | 0.91 | 3.95 (0.150) | 73.2 | 1.90 | |
| Huber | 0.77 (0.025) | 86.4 | 0.55 | 0.94 (0.032) | 84.5 | 0.69 | 1.94 (0.071) | 79.3 | 1.19 | |
| 400 | LS | 0.80 (0.026) | 86.0 | 0.58 | 2.69 (0.136) | 77.8 | 1.34 | 54.7 | 4.26 | |
| p(0.5) | 0.39 (0.013) | 90.5 | 0.27 | 0.44 (0.017) | 89.6 | 0.32 | 0.71 (0.024) | 86.9 | 0.50 | |
| p(0.25) | 0.56 (0.019) | 88.8 | 0.37 | 0.66 (0.020) | 86.9 | 0.50 | 1.70 (0.055) | 79.6 | 1.17 | |
| Huber | 0.38 (0.012) | 90.4 | 0.27 | 0.48 (0.017) | 88.8 | 0.36 | 0.91 (0.029) | 84.9 | 0.65 | |
| 800 | LS | 0.41 (0.013) | 89.9 | 0.29 | 1.35 (0.150) | 83.1 | 0.82 | 56.5 | 4.00 | |
| p(0.5) | 0.18 (0.006) | 93.6 | 0.12 | 0.20 (0.007) | 92.6 | 0.16 | 0.36 (0.013) | 91.0 | 0.25 | |
| p(0.25) | 0.28 (0.009) | 92.2 | 0.18 | 0.31 (0.010) | 90.8 | 0.24 | 0.89 (0.031) | 85.8 | 0.60 | |
| Huber | 0.19 (0.006) | 93.3 | 0.13 | 0.22 (0.007) | 92.1 | 0.18 | 0.47 (0.017) | 89.2 | 0.34 | |
LS stands for lsA-learning. P(0.5) stands for robust regression with pinball loss and parameter τ = 0.5. P(0.25) stands for robust regression with pinball loss and parameter τ = 0.25. Huber stands for robust regression with Huber loss, where parameter α is tuned automatically with R function rlm. Column δ0 5 is multiplied by 10.