Table 2.
Bias and variances for the change-point estimate τ̂n compared for one change-point problem in 3 setups: A: θT = (0.2, 1, 1, 0.6), B: θT = (0.3, 1.5, 1, 0.8) & C: θT = (0.3, 1.5,−1, 0.2) (S.D.: Average of estimated standard deviations over 1000 replications; Emp. S.D. : Sample standard deviation based on 1000 replications).
| Sample | Bias (S.D., Emp. S.D.) | Theoretical | |
|---|---|---|---|
| Size | × 10−3 | S.D. | |
| n | Existing | Proposed | × 10−3 |
| Set-up A | |||
| 50 | −8.2 (58.1, 60.3) | −12.4 (58.3, 61.0) | 57.8 |
| 100 | −4.3 (41.0, 41.2) | −5.2 (41.0, 41.3) | 40.9 |
| 500 | −2.1 (18.3, 18.4) | −2.9 (18.3, 18.5) | 18.3 |
| 1000 | −0.6 (12.9, 12.9) | −0.9 (12.9, 13.0) | 12.9 |
| 5000 | −0.0 (5.8, 5.8) | −0.1 (5.8, 5.8) | 5.8 |
| Set-up B | |||
| 50 | −9.2 (71.6, 73.2) | −14.1 (71.6, 73.6) | 70.7 |
| 100 | −4.7 (50.1, 50.3) | −5.9 (50.1, 50.4) | 50.0 |
| 500 | −2.8 (22.4, 22.4) | −3.3 (22.4, 22.5) | 22.3 |
| 1000 | −0.9 (15.8, 15.8) | −1.2 (15.8, 15.8) | 15.8 |
| 5000 | −0.1 (7.1, 7.1) | −0.1 (7.1, 7.1) | 7.1 |
| Set-up C | |||
| 50 | 8.1 (71.6, 73.3) | 12.8 (71.6, 73.5) | 70.7 |
| 100 | 4.7 (50.1, 50.4) | 6.2 (50.1, 50.5) | 50.0 |
| 500 | 1.8 (22.4, 22.4) | 2.0 (22.5, 22.6) | 22.3 |
| 1000 | 0.2 (15.8, 15.8) | 0.4 (15.8, 15.8) | 15.8 |
| 5000 | 0.1 (7.1, 7.1) | 0.1 (7.1, 7.1) | 7.1 |