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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