Figure 1.
Monte Carlo results for the contaminated simple regression model. Mean square estimation error MSEE() for varying and . CUBIF11r, red; RQL01r, green; MT-Ir, blue; MD10r, black.
Figure 1.
Monte Carlo results for the contaminated simple regression model. Mean square estimation error MSEE() for varying and . CUBIF11r, red; RQL01r, green; MT-Ir, blue; MD10r, black.
Figure 2.
Monte Carlo results for the contaminated simple regression model. Mean square estimation error MSEE() for varying and . RQL14, green; MT23, blue; MD04, orange; MCML10, black.
Figure 2.
Monte Carlo results for the contaminated simple regression model. Mean square estimation error MSEE() for varying and . RQL14, green; MT23, blue; MD04, orange; MCML10, black.
Figure 3.
Efficiency of MD07 (blue) and MD10 (black) as a function of (solid) and of (broken): , , ; the solid line is obtained with and ; the broken line is obtained with , and .
Figure 3.
Efficiency of MD07 (blue) and MD10 (black) as a function of (solid) and of (broken): , , ; the solid line is obtained with and ; the broken line is obtained with , and .
Figure 4.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Starting estimates: MD07r, light green; MD10r, cyan; MCML07, grey; MCML10, black; (ML, orange).
Figure 4.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Starting estimates: MD07r, light green; MD10r, cyan; MCML07, grey; MCML10, black; (ML, orange).
Figure 5.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Simple DCML estimates: MD07r + D, light green; MD10r + D, cyan; MCML07 + D, grey; MCML10 + D, black.
Figure 5.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Simple DCML estimates: MD07r + D, light green; MD10r + D, cyan; MCML07 + D, grey; MCML10 + D, black.
Figure 6.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Iterated DCML estimates: MCML++, pink; MCML*+, red; (MCML10 + D, black).
Figure 6.
Monte Carlo results for the contaminated multiple regression model. Mean square estimation error MSEE() for varying and . Iterated DCML estimates: MCML++, pink; MCML*+, red; (MCML10 + D, black).
Figure 7.
Boxplots of the coefficient estimates , , , and of the NB model for length of stays (); A = ML, B = MD10r, C = MCML10, D = MCML + D.
Figure 7.
Boxplots of the coefficient estimates , , , and of the NB model for length of stays (); A = ML, B = MD10r, C = MCML10, D = MCML + D.
Table 1.
Efficiencies of the robust Poisson regression candidates: Eff is the empirical efficiency based on complete data; Effr is the empirical efficiency based on cleaned data; Eff is the asymptotic efficiency. CUBIF, conditionally unbiased bounded influence estimator; RQL, robust quasi-likelihood estimator; MT, transformed M estimator; MD, minimum density power divergence estimator; MCML, modified conditionally maximum likelihood estimator.
Table 1.
Efficiencies of the robust Poisson regression candidates: Eff is the empirical efficiency based on complete data; Effr is the empirical efficiency based on cleaned data; Eff is the asymptotic efficiency. CUBIF, conditionally unbiased bounded influence estimator; RQL, robust quasi-likelihood estimator; MT, transformed M estimator; MD, minimum density power divergence estimator; MCML, modified conditionally maximum likelihood estimator.
Estimator | Eff | Effr | Eff |
---|
CUBIF11 | 0.62 | 0.52 | 0.66 |
CUBIF18 | 0.89 | 0.73 | 0.92 |
RQL01 | 0.68 | 0.54 | 0.66 |
RQL14 | 0.94 | 0.75 | 0.93 |
MT-I | 0.85 | 0.78 | —- |
MT23 | 0.87 | 0.70 | 0.90 |
MD04 | 0.84 | 0.67 | 0.89 |
MD07 | 0.75 | 0.60 | 0.79 |
MD10 | 0.66 | 0.54 | 0.68 |
MCML04 | 0.78 | —- | 1.00 |
MCML07 | 0.78 | —- | 1.00 |
MCML10 | 0.78 | —- | 1.00 |
Table 2.
Monte Carlo efficiencies (%) of the MDE and MCML estimators of the Poisson regression for , , .
Table 2.
Monte Carlo efficiencies (%) of the MDE and MCML estimators of the Poisson regression for , , .
p | n | | MDE07r | MDE10r | MCML07 | MCML10 | MCML+ | MCML* |
---|
5 | 25 | | 26 | 22 | 36 | 32 | 54 | 62 |
5 | 25 | | 61 | 47 | 67 | 59 | 80 | 87 |
5 | 25 | | 83 | 66 | 83 | 76 | 88 | 93 |
5 | 25 | | 88 | 73 | 89 | 82 | 91 | 94 |
10 | 50 | | 22 | 18 | 42 | 38 | 56 | 69 |
10 | 50 | | 54 | 39 | 72 | 64 | 79 | 90 |
10 | 50 | | 71 | 51 | 82 | 73 | 85 | 93 |
10 | 50 | | 77 | 56 | 85 | 77 | 87 | 94 |
20 | 100 | | 22 | 19 | 57 | 55 | 73 | 84 |
20 | 100 | | 54 | 45 | 86 | 84 | 94 | 98 |
20 | 100 | | 68 | 56 | 91 | 89 | 97 | 99 |
20 | 100 | | 74 | 61 | 93 | 91 | 97 | 99 |
Table 3.
Monte Carlo efficiencies (%) of the MDE and MCML estimators of the Poisson regression for , , .
Table 3.
Monte Carlo efficiencies (%) of the MDE and MCML estimators of the Poisson regression for , , .
p | n | | MDE07r | MDE10r | MCML07 | MCML10 | MCML+ | MCML* |
---|
5 | 25 | | 47 | 42 | 58 | 55 | 78 | 83 |
5 | 25 | | 73 | 72 | 84 | 81 | 95 | 97 |
5 | 25 | | 85 | 87 | 93 | 91 | 98 | 99 |
5 | 25 | | 90 | 91 | 96 | 94 | 99 | 99 |
10 | 50 | | 40 | 35 | 67 | 65 | 82 | 87 |
10 | 50 | | 67 | 66 | 91 | 89 | 97 | 99 |
10 | 50 | | 78 | 79 | 96 | 94 | 99 | 100 |
10 | 50 | | 83 | 84 | 97 | 95 | 99 | 100 |
20 | 100 | | 39 | 37 | 79 | 78 | 89 | 93 |
20 | 100 | | 65 | 69 | 96 | 95 | 99 | 100 |
20 | 100 | | 73 | 79 | 98 | 97 | 100 | 100 |
20 | 100 | | 77 | 93 | 98 | 98 | 100 | 100 |
Table 4.
Monte Carlo maximum MSEE (over ) of the starting and DCML estimators of the Poisson regression under point contamination in , where (, , , ).
Table 4.
Monte Carlo maximum MSEE (over ) of the starting and DCML estimators of the Poisson regression under point contamination in , where (, , , ).
| | | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|
MDE07r | 0.12 |
0.10 | 0.12 | 0.14 | 0.16 | 0.15 | 0.16 | 0.14 | 0.12 |
MDE10r | 0.12 |
0.13 | 0.12 | 0.15 | 0.15 | 0.16 | 0.17 | 0.16 | 0.12 |
MCML07 | 0.11 |
0.09 | 0.10 | 0.13 | 0.14 | 0.14 | 0.14 | 0.13 | 0.11 |
MCML10 | 0.11 |
0.09 | 0.10 | 0.13 | 0.13 | 0.14 | 0.14 | 0.13 | 0.11 |
MDE07r + D | 0.11 |
0.08 | 0.08 | 0.10 | 0.13 | 0.12 | 0.14 | 0.13 | 0.10 |
MDE10r + D | 0.13 |
0.10 | 0.09 | 0.13 | 0.12 | 0.13 | 0.13 | 0.13 | 0.11 |
MCML07 + D | 0.09 |
0.06 | 0.07 | 0.08 | 0.12 | 0.11 | 0.13 | 0.12 | 0.09 |
MCML10 + D | 0.08 |
0.06 | 0.07 | 0.09 | 0.10 | 0.10 | 0.11 | 0.11 | 0.09 |
MCML++ | 0.09 |
0.06 | 0.06 | 0.08 | 0.11 | 0.11 | 0.11 | 0.12 | 0.09 |
MCML*+ | 0.10 |
0.06 | 0.06 | 0.08 | 0.12 | 0.11 | 0.11 | 0.12 | 0.09 |
Table 5.
Bootstrap coefficient means and standard errors of the ML, MDE10r, MCML10, and MCML10 + D estimators of the negative binomial (NB) regression for length of stay data (); var is the total variance; rej is the number of rejected samples.
Table 5.
Bootstrap coefficient means and standard errors of the ML, MDE10r, MCML10, and MCML10 + D estimators of the negative binomial (NB) regression for length of stay data (); var is the total variance; rej is the number of rejected samples.
| | | | | | | | var | rej |
---|
ML | 2.07 | 0.02 | 0.03 | −0.01 | 0.09 | 0.00 | 0.88 | 0.49 | 0 |
| (0.47) | (0.03) | (0.02) | (0.01) | (0.52) | (0.01) | | | |
MD10r | 1.56 | 0.02 | 0.06 | −0.01 | 0.42 | 0.01 | 0.67 | 0.85 | 0 |
| (0.54) | (0.05) | (0.04) | (0.01) | (0.74) | (0.02) | | | |
MCML10 | 1.70 | 0.02 | 0.05 | −0.01 | 0.29 | 0.01) | 0.48 | 0.51 | 1 |
| (0.43) | (0.04) | (0.04) | (0.01) | (0.57) | (0.02) | | | |
MCML10 + D | 1.95 | 0.02 | 0.04 | −0.01 | 0.15 | 0.01 | 0.79 | 0.39 | 45 |
| (0.45) | (0.03) | (0.02) | (0.01) | (0.50) | (0.01) | | | |
Table 6.
Bootstrap coefficient means and standard errors of the ML, MDE10r, MCML10, and MCML10 + D estimators of the NB regression for length of stay data (); var is the total variance; rej is the number of rejected samples.
Table 6.
Bootstrap coefficient means and standard errors of the ML, MDE10r, MCML10, and MCML10 + D estimators of the NB regression for length of stay data (); var is the total variance; rej is the number of rejected samples.
| | | | | | var | rej |
---|
ML | 2.04 | 0.04 | 0.02 | 0.00 | 0.93 | 0.41 | 8 |
| (0.64) | (0.04) | (0.03) | (0.01) | | | |
MD10r | 1.59 | 0.04 | 0.02 | 0.00 | 0.85 | 0.99 | 0 |
| (0.99) | (0.08) | (0.07) | (0.02) | | | |
MCML10 | 1.72 | 0.04 | 0.04 | 0.00 | 0.52 | 1.70 | 2 |
| (1.21) | (0.35) | (0.31) | (0.14) | | | |
MCML10 + D | 1.92 | 0.03 | 0.02 | 0.00 | 0.86 | 0.46 | 8 |
| (0.67) | (0.05) | (0.04) | (0.01) | | | |