Table 2 Data analysis results for the second real data set

\(\ell\)

− 250.1338

− 255.7655

− 252.0314

− 255.1664

− 251.4986

− 2\(\ell\)

500.2676

511.5310

504.0628

510.3328

502.9972

AIC

506.2675

515.5528

510.0627

512.3328

506.9973

BIC

513.0975

520.1062

516.8927

514.6095

511.5506

CAIC

506.6205

515.7267

510.4157

512.3900

507.1712

HQIC

508.9866

517.3655

512.7818

513.2392

508.8100

K-S

0.0870

0.2030

0.1019

0.1969

0.1054

A-D

0.5465

3.8577

0.9028

3.4854

0.8450

CvM

0.0944

0.5319

0.1683

0.4856

0.1488

p-value (K-S)

0.6464

0.0053

0.4427

0.0075

0.4004

p-value (A-D)

0.6994

0.0103

0.4121

0.0157

0.4492

p-value (CvM)

0.6146

0.0328

0.3389

0.0431

0.3939

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

1.6567

0.1681

0.0556

0.1082

0.9011

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

0.0796

0.1116

0.9727

–

0.1096

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

0.0261

–

0.9985

–

–

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

1.5216

0.5792

0.0678

0.0109

0.0855

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

0.0142

0.0156

0.4156

–

0.0301

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

0.0407

–

0.3020

–

–

Numerical Method

NM

CG

NM

CG

BFGS

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

52.0855

0.8000

14.0013

2.6950

0.6077

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

0.5705

0.8000

16.1198

–

37.2225

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

42.8379

–

0.6379

–

–