Table 1 Data analysis results for the first real data set

\(\ell\)

− 46.5507

− 50.0825

− 51.2945

− 95.0418

− 50.0752

− 2\(\ell\)

93.1014

100.1650

102.5890

190.0836

100.1503

AIC

99.1014

104.1650

108.5890

192.0836

104.1503

BIC

104.8375

107.9890

114.3251

193.9956

107.9743

CAIC

99.6232

104.4203

109.1108

192.1669

104.4056

HQIC

101.2858

105.6212

110.7734

192.8117

105.6065

K-S

0.0712

0.1141

0.1200

0.4432

0.1299

A-D

0.2208

1.0201

1.1767

13.8602

0.9453

CvM

0.0273

0.1609

0.1767

2.9365

0.1453

p-value (K-S)

0.9618

0.5336

0.4677

0.0000

0.3675

p-value (A-D)

0.9836

0.3465

0.2764

0.0000

0.3868

p-value (CvM)

0.9849

0.3589

0.3182

0.0000

0.4053

\({\widehat{p}}_{1}\)

8.6078

42.1315

1.3734

0.5184

4.7833

\({\widehat{p}}_{2}\)

2.7786

1.6689

55.6645

–

0.0038

\({\widehat{p}}_{3}\)

1355.9139

–

0.9999

–

–

\(SE({{\widehat{p}}_{1}})\)

2.3069

16.5562

1.3374

0.0569

0.2501

\(SE({{\widehat{p}}_{2}})\)

0.5230

0.1597

99.4232

–

0.0011

\(SE({{\widehat{p}}_{3}})\)

2874.8663

–

0.4989

–

–

Numerical Methods

L-BFGS-B

L-BFGS-B

BFGS

CG

NM

Initial value for \({\widehat{p}}_{1}\)

96.2858

12.0576

60.6673

46.2549

1.0779

Initial value for \({\widehat{p}}_{2}\)

4.5543

0.2161

87.5668

–

3.6835

Initial value for \({\widehat{p}}_{3}\)

79.4243

–

0.9679

–

–