Table 3 MLE and Bayesian for TCPE parameters based on Type-II UPHCS: \(\lambda =0.7, \zeta =1.4, \eta =1.6\).

1.2,1.6

1

0.6

\(\lambda\)

\(-\)0.3243

0.1476

\(-\)0.0572

0.0084

\(-\)0.0560

0.0082

\(-\)0.0583

0.0085

0.8081

96.29%

0.0293

0.0294

0.2746

0.2741

0.2744

\(\zeta\)

0.7136

0.7225

0.0265

0.0187

0.0292

0.0190

0.0238

0.0185

1.8119

96.14%

0.0695

0.0692

0.5032

0.5078

0.5053

\(\eta\)

\(-\)0.3869

0.2620

0.0253

0.0158

0.0276

0.0160

0.0230

0.0156

1.3146

95.14%

0.0508

0.0509

0.4712

0.4686

0.4687

0.8

\(\lambda\)

\(-\)0.2868

0.1071

\(-\)0.0385

0.0050

\(-\)0.0377

0.0049

\(-\)0.0394

0.0051

0.6189

95.71%

0.0239

0.0239

0.2358

0.2363

0.2353

\(\zeta\)

0.5558

0.4425

0.0119

0.0108

0.0134

0.0109

0.0104

0.0108

1.4337

96.00%

0.0557

0.0555

0.3981

0.3983

0.3971

\(\eta\)

\(-\)0.3356

0.1928

0.0219

0.0103

0.0233

0.0103

0.0204

0.0102

1.1111

96.00%

0.0444

0.0432

0.3788

0.3788

0.3780

2

0.6

\(\lambda\)

\(-\)0.1230

0.0780

\(-\)0.0212

0.0071

\(-\)0.0199

0.0071

\(-\)0.0226

0.0071

0.9834

96.00%

0.0364

0.0363

0.3073

0.3064

0.3073

\(\zeta\)

0.2782

0.2698

0.0179

0.0157

0.0203

0.0158

0.0156

0.0155

1.7207

94.43%

0.0655

0.0649

0.4792

0.4817

0.4787

\(\eta\)

\(-\)0.1060

0.2114

0.0049

0.0170

0.0073

0.0171

0.0025

0.0169

1.7550

95.29%

0.0673

0.0672

0.5090

0.5045

0.5084

0.8

\(\lambda\)

\(-\)0.2159

0.0709

\(-\)0.0254

0.0042

\(-\)0.0245

0.0041

\(-\)0.0263

0.0042

0.7077

95.86%

0.0267

0.0268

0.2264

0.2260

0.2270

\(\zeta\)

0.1401

0.1827

0.0057

0.0104

0.0072

0.0105

0.0042

0.0104

1.3710

96.29%

0.0515

0.0516

0.3800

0.3834

0.3773

\(\eta\)

\(-\)0.2375

0.1657

0.0168

0.0096

0.0183

0.0097

0.0153

0.0095

1.2972

96.29%

0.0491

0.0496

0.3562

0.3571

0.3559

3

0.6

\(\lambda\)

\(-\)0.2528

0.1088

\(-\)0.0421

0.0070

\(-\)0.0409

0.0069

\(-\)0.0433

0.0071

0.8315

96.00%

0.0312

0.0317

0.2857

0.2855

0.2855

\(\zeta\)

0.5380

0.4975

0.0224

0.0182

0.0251

0.0184

0.0198

0.0179

1.7894

95.43%

0.0646

0.0661

0.5357

0.5329

0.5336

\(\eta\)

\(-\)0.2873

0.2082

0.0212

0.0154

0.0236

0.0156

0.0188

0.0152

1.3907

96.14%

0.0525

0.0506

0.4679

0.4666

0.4680

0.8

\(\lambda\)

\(-\)0.2570

0.0960

\(-\)0.0354

0.0048

\(-\)0.0345

0.0047

\(-\)0.0362

0.0048

0.6791

97.14%

0.0267

0.0265

0.2270

0.2271

0.2268

\(\zeta\)

0.4930

0.3903

0.0099

0.0108

0.0115

0.0108

0.0083

0.0107

1.5054

95.86%

0.0554

0.0558

0.4129

0.4135

0.4097

\(\eta\)

\(-\)0.2982

0.1865

0.0168

0.0105

0.0183

0.0106

0.0152

0.0104

1.2254

96.86%

0.0453

0.0455

0.3855

0.3854

0.3853

0.75,1.5

1

0.6

\(\lambda\)

\(-\)0.2848

0.1608

\(-\)0.0547

0.0086

\(-\)0.0536

0.0085

\(-\)0.0558

0.0087

1.1075

91.57%

0.0389

0.0403

0.2815

0.2812

0.2818

\(\zeta\)

0.6533

0.7176

0.0242

0.0172

0.0267

0.0175

0.0217

0.0170

2.1154

94.71%

0.0792

0.0792

0.5118

0.5107

0.5121

\(\eta\)

\(-\)0.3320

0.3192

0.0236

0.0172

0.0259

0.0174

0.0212

0.0170

1.7932

93.29%

0.0700

0.0702

0.4813

0.4820

0.4799

0.8

\(\lambda\)

\(-\)0.3357

0.1359

\(-\)0.0483

0.0058

\(-\)0.0475

0.0057

\(-\)0.0491

0.0059

0.5972

94.43%

0.0219

0.0219

0.2245

0.2246

0.2246

\(\zeta\)

0.6658

0.5943

0.0120

0.0108

0.0135

0.0109

0.0105

0.0108

1.5248

96.71%

0.0594

0.0590

0.4160

0.4175

0.4145

\(\eta\)

\(-\)0.4111

0.2411

0.0211

0.0112

0.0226

0.0113

0.0196

0.0111

1.0536

94.86%

0.0399

0.0398

0.4196

0.4198

0.4171

2

0.6

\(\lambda\)

\(-\)0.0777

0.0766

\(-\)0.0140

0.0067

\(-\)0.0126

0.0067

\(-\)0.0153

0.0067

1.0419

96.00%

0.0391

0.0389

0.3037

0.3040

0.3026

\(\zeta\)

0.2135

0.2320

0.0139

0.0160

0.0163

0.0161

0.0115

0.0158

1.6937

94.43%

0.0667

0.0677

0.4921

0.4897

0.4902

\(\eta\)

\(-\)0.0523

0.2204

0.0004

0.0158

0.0028

0.0159

\(-\)0.0020

0.0157

1.8300

95.00%

0.0678

0.0677

0.4827

0.4851

0.4811

0.8

\(\lambda\)

\(-\)0.0623

0.0692

\(-\)0.0296

0.0045

\(-\)0.0287

0.0045

\(-\)0.0304

0.0046

0.7664

96.00%

0.0290

0.0288

0.2305

0.2307

0.2307

\(\zeta\)

0.1942

0.2131

0.0021

0.0097

0.0037

0.0097

0.0006

0.0096

1.4548

95.57%

0.0560

0.0549

0.3699

0.3707

0.3694

\(\eta\)

\(-\)0.0427

0.1942

0.0131

0.0110

0.0147

0.0110

0.0116

0.0109

1.3667

96.57%

0.0519

0.0524

0.3925

0.3911

0.3956

3

0.6

\(\lambda\)

\(-\)0.2062

0.1164

\(-\)0.0347

0.0073

\(-\)0.0335

0.0072

\(-\)0.0359

0.0074

1.0664

93.29%

0.0402

0.0401

0.2986

0.3004

0.2982

\(\zeta\)

0.4654

0.4880

0.0070

0.0166

0.0095

0.0167

0.0045

0.0165

2.0435

97.43%

0.0762

0.0765

0.5094

0.5093

0.5050

\(\eta\)

\(-\)0.2310

0.2475

0.0107

0.0162

0.0131

0.0164

0.0082

0.0160

1.7286

94.86%

0.0636

0.0635

0.4854

0.4880

0.4870

0.8

\(\lambda\)

\(-\)0.2933

0.1156

\(-\)0.0407

0.0049

\(-\)0.0399

0.0049

\(-\)0.0415

0.0050

0.6739

95.43%

0.0251

0.0250

0.2220

0.2214

0.2221

\(\zeta\)

0.5590

0.4657

0.0099

0.0103

0.0114

0.0103

0.0084

0.0102

1.5355

96.71%

0.0590

0.0593

0.3919

0.3938

0.3899

\(\eta\)

\(-\)0.3531

0.2183

0.0214

0.0109

0.0229

0.0110

0.0199

0.0108

1.2002

96.14%

0.0448

0.0447

0.3956

0.3959

0.3930