Table 5 The AEs and MSE obtained from various estimation methods for the NLxEx distribution.

From: The flexible Lomax-G family with estimation methods and applications in hydrology and biomedicine

n

ML

LS

WLS

CVM

AD

RTAD

Set 10: \((\alpha =0.2, \beta =1.4, \lambda =2.5)\)

AEs MSE

AEs MSE

AEs MSE

AEs MSE

AEs MSE

AEs MSE

50

0.781 0.241

0.947 0.374

0.936 0.361

0.923 0.344

0.931 0.361

0.911 0.327

 

2.142 0.487

2.674 0.738

2.652 0.725

2.584 0.716

2.557 0.694

2.538 0.636

 

3.147 0.751

3.754 0.974

3.738 0.946

3.721 0.931

3.715 0.906

3.698 0.875

100

0.572 0.223

0.913 0.348

0.911 0.331

0.901 0.326

0.895 0.313

0.876 0.301

 

2.021 0.442

2.543 0.716

2.463 0.701

2.387 0.685

2.372 0.653

2.361 0.614

 

3.104 0.725

3.583 0.959

3.556 0.934

3.531 0.920

3.510 0.887

3.498 0.852

200

0.461 0.211

0.842 0.326

0.837 0.318

0.821 0.313

0.811 0.301

0.797 0.287

 

1.854 0.421

2.385 0.574

2.361 0.566

2.356 0.541

2.339 0.530

2.314 0.511

 

2.971 0.614

3.517 0.914

3.496 0.911

3.452 0.878

3.426 0.854

3.411 0.829

500

0.328 0.186

0.637 0.313

0.615 0.310

0.602 0.286

0.597 0.289

0.567 0.254

 

1.613 0.402

2.352 0.543

2.347 0.531

2.332 0.511

2.313 0.501

2.287 0.500

 

2.680 0.507

3.301 0.785

3.294 0.761

3.275 0.748

3.255 0.731

3.236 0.712

750

0.210 0.153

0.475 0.286

0.468 0.271

0.456 0.254

0.449 0.248

0.438 0.236

 

1.403 0.321

2.196 0.536

2.177 0.518

2.131 0.487

2.120 0.463

1.975 0.411

 

2.511 0.410

3.264 0.755

3.227 0.742

3.208 0.721

3.164 0.711

3.014 0.678

Set 11: \((\alpha =2, \beta =2.2, \lambda =1)\)

50

2.712 0.321

2.975 0.635

2.958 0.621

2.937 0.601

2.913 0.587

2.900 0.554

 

2.957 0.542

3.424 0.763

3.385 0.748

3.365 0.726

3.348 0.714

3.316 0.702

 

1.557 0.286

1.963 0.525

1.955 0.517

1.936 0.506

1.915 0.487

1.902 0.465

100

2.483 0.284

2.764 0.617

2.752 0.611

2.736 0.585

2.718 0.553

2.702 0.521

 

2.774 0.446

3.285 0.672

3.269 0.656

3.228 0.643

3.212 0.625

3.164 0.614

 

1.414 0.257

1.765 0.511

1.752 0.501

1.743 0.498

1.728 0.471

1.711 0.447

200

2.401 0.245

2.741 0.575

2.720 0.542

2.693 0.518

2.665 0.497

2.644 0.462

 

2.543 0.418

3.164 0.644

3.139 0.627

3.112 0.604

2.975 0.587

2.795 0.556

 

1.217 0.229

1.738 0.487

1.711 0.449

1.688 0.416

1.643 0.401

1.612 0.387

500

2.256 0.161

2.615 0.548

2.602 0.536

2.578 0.513

2.543 0.452

2.524 0.422

 

2.413 0.313

3.023 0.515

2.977 0.487

2.823 0.455

2.756 0.431

2.716 0.410

 

1.174 0.206

1.715 0.453

1.687 0.436

1.656 0.414

1.631 0.387

1.602 0.341

750

2.011 0.072

2.423 0.514

2.401 0.489

2.236 0.461

2.211 0.432

2.198 0.396

 

2.212 0.207

2.976 0.489

2.944 0.466

2.861 0.421

2.562 0.387

2.529 0.374

 

1.013 0.115

1.585 0.431

1.477 0.430

1.471 0.401

1.310 0.354

1.176 0.311

Set 12: \((\alpha =1.6, \beta =1.1, \lambda =0.6)\)

50

1.842 0.236

2.041 0.362

2.003 0.341

1.924 0.328

1.978 0.352

1.895 0.301

 

1.287 0.183

1.435 0.321

1.418 0.332

1.401 0.306

1.392 0.314

1.364 0.276

 

0.783 0.145

0.874 0.298

0.863 0.274

0.851 0.266

0.872 0.281

0.826 0.251

100

1.731 0.184

1.932 0.311

1.901 0.298

1.874 0.286

1.862 0.295

1.812 0.241

 

1.214 0.132

1.318 0.247

1.296 0.236

1.284 0.228

1.277 0.236

1.251 0.205

 

0.723 0.112

0.804 0.241

0.796 0.221

0.781 0.215

0.792 0.219

0.766 0.198

200

1.664 0.116

1.784 0.224

1.756 0.212

1.743 0.205

1.731 0.213

1.712 0.194

 

1.157 0.074

1.235 0.186

1.221 0.171

1.214 0.168

1.212 0.174

1.196 0.162

 

0.671 0.086

0.732 0.185

0.721 0.172

0.719 0.168

0.726 0.171

0.704 0.159

500

1.612 0.061

1.683 0.162

1.674 0.156

1.662 0.151

1.659 0.154

1.652 0.143

 

1.112 0.041

1.176 0.134

1.161 0.128

1.158 0.124

1.156 0.129

1.149 0.119

 

0.632 0.052

0.671 0.152

0.666 0.141

0.661 0.138

0.662 0.141

0.656 0.132

750

1.598 0.032

1.649 0.141

1.646 0.136

1.641 0.133

1.639 0.135

1.637 0.129

 

1.101 0.023

1.152 0.121

1.148 0.117

1.144 0.115

1.143 0.118

1.138 0.112

 

0.613 0.028

0.642 0.138

0.639 0.132

0.636 0.131

0.637 0.133

0.634 0.126

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