From: The generalized Cauchy family of distributions with applications
Sample size | Actual values | ML estimates | Standard deviation | ||||||
---|---|---|---|---|---|---|---|---|---|
n | α | β | \( \theta \) | \( \widehat{\alpha} \) | \( \widehat{\beta} \) | \( \widehat{\theta} \) | \( \widehat{\alpha} \) | \( \widehat{\beta} \) | \( \widehat{\theta} \) |
50 | 1 | 1 | 1 | 1.1605 | 0.9260 | 1.1455 | 0.0518 | 0.0460 | 0.0538 |
0.5 | 1 | 2 | 0.5377 | 0.9552 | 2.1399 | 0.0180 | 0.0426 | 0.0978 | |
0.5 | 3 | 2 | 0.5351 | 2.8849 | 2.1952 | 0.0154 | 0.1207 | 0.1065 | |
0.9 | 0.5 | 1 | 1.0229 | 0.4960 | 1.0579 | 0.0481 | 0.0222 | 0.0442 | |
2 | 0.5 | 1 | 2.9571 | 0.4448 | 1.2255 | 0.4271 | 0.0282 | 0.0917 | |
5 | 0.5 | 1 | 7.9697 | 0.4487 | 0.8862 | 0.9537 | 0.0183 | 0.0737 | |
100 | 1 | 1 | 1 | 1.0717 | 0.9617 | 1.0499 | 0.0216 | 0.0230 | 0.0248 |
0.5 | 1 | 2 | 0.5182 | 1.0090 | 2.0843 | 0.0084 | 0.0229 | 0.0457 | |
0.5 | 3 | 2 | 0.5165 | 2.9594 | 2.0505 | 0.0065 | 0.0591 | 0.0441 | |
0.9 | 0.5 | 1 | 0.9375 | 0.4932 | 1.0309 | 0.0181 | 0.0105 | 0.0203 | |
2 | 0.5 | 1 | 2.2258 | 0.4753 | 1.0866 | 0.0646 | 0.0131 | 0.0281 | |
5 | 0.5 | 1 | 6.3436 | 0.4737 | 0.9243 | 0.4015 | 0.0089 | 0.0367 | |
300 | 1 | 1 | 1 | 1.0244 | 0.9983 | 1.0270 | 0.0094 | 0.0114 | 0.0111 |
0.5 | 1 | 2 | 0.5076 | 0.9908 | 2.0320 | 0.0041 | 0.0102 | 0.0229 | |
0.5 | 3 | 2 | 0.5113 | 2.9636 | 2.0718 | 0.0039 | 0.0315 | 0.0241 | |
0.9 | 0.5 | 1 | 0.9242 | 0.4953 | 1.0159 | 0.0086 | 0.0053 | 0.0102 | |
2 | 0.5 | 1 | 2.1286 | 0.4844 | 1.0369 | 0.0270 | 0.0071 | 0.0123 | |
5 | 0.5 | 1 | 5.2815 | 0.4987 | 0.9562 | 0.1368 | 0.0040 | 0.0149 |