Table 3 Average bias (standard deviation) for the MLEs
From: Families of distributions arising from the quantile of generalized lambda distribution
Actual values | Average bias | Mode(s) | |||||
|---|---|---|---|---|---|---|---|
λ 3 | λ 4 | θ | n | \( {\widehat{\lambda}}_3 \) | \( {\widehat{\lambda}}_4 \) | \( \widehat{\theta} \) | |
0.8 | 0.6 | 0.5 | 50 | -0.0120 (0.1765) | 0.0332 (0.1666) | -0.0405 (0.1156) | Reversed J-shape with one mode at x = 0 |
100 | -0.0415 (0.1498) | 0.0382 (0.1565) | -0.0495 (0.1092) | ||||
250 | -0.0357 (0.1100) | 0.0294 (0.1348) | -0.0385 (0.0944) | ||||
500 | -0.0240 (0.0828) | 0.0127 (0.1216) | -0.0238 (0.0807) | ||||
1000 | -0.0130 (0.0591) | -0.0000 (0.1081) | -0.0121 (0.0658) | ||||
1 | 2 | 2 | 50 | -0.0289 (0.2693) | 0.1108 (0.5991) | -0.1246 (0.4180) | Unimodal at x = 0 |
100 | -0.0336 (0.2405) | 0.0757 (0.5425) | -0.1135 (0.3728) | ||||
250 | -0.0345 (0.2089) | 0.0496 (0.4811) | -0.0942 (0.2870) | ||||
500 | -0.0284 (0.1839) | 0.0436 (0.4072) | -0.0713 (0.2122) | ||||
1000 | -0.0469 (0.1769) | -0.0014 (0.3396) | -0.0549 (0.1534) | ||||
2 | 0.8 | 1 | 50 | 0.1337 (0.5625) | 0.0237 (0.2040) | -0.0222 (0.2059) | Right-skewed with one mode at x > 0 |
100 | 0.0073 (0.5240) | 0.0424 (0.1978) | -0.0504 (0.1845) | ||||
250 | -0.1297 (0.4199) | 0.0381 (0.1895) | -0.0684 (0.1531) | ||||
500 | -0.1440 (0.3393) | 0.0170 (0.1692) | -0.0506 (0.1224) | ||||
1000 | -0.1121 (0.2746) | -0.0019 (0.1484) | -0.0281 (0.0925) | ||||
3 | 0.5 | 3 | 50 | -0.0571 (0.7754) | -0.0030 (0.1311) | -0.0908 (0.5682) | Right-skewed with one mode at x > 0 |
100 | -0.0851 (0.7237) | -0.0018 (0.1190) | -0.0870 (0.5159) | ||||
250 | -0.1058 (0.6366) | -0.0054 (0.0990) | -0.0529 (0.4076) | ||||
500 | -0.1073 (0.5306) | -0.0088 (0.0836) | -0.0309 (0.3365) | ||||
1000 | -0.0796 (0.4066) | -0.0034 (0.0632) | -0.0256 (0.2526) | ||||
4 | 0.7 | 2 | 50 | 0.0921 (1.0819) | 0.0284 (0.1753) | -0.0651 (0.3435) | Bimodal with two modes at x = 0 and x > 0 |
100 | -0.0580 (1.0216) | 0.0110 (0.1722) | -0.0609 (0.3031) | ||||
250 | -0.1211 (0.8777) | -0.0007 (0.1428) | -0.0393 (0.2348) | ||||
500 | -0.1794 (0.7393) | -0.0030 (0.1183) | -0.0360 (0.1914) | ||||
1000 | -0.1301 (0.5861) | 0.0010 (0.0868) | -0.0256 (0.1454) | ||||