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Table 3 Estimates and standard deviations for the parameters using MLE method

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