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Table 12 Performance of tolerance intervals based on normal distribution when the true distribution is Cauchy (G: Normal; F: Cauchy)

From: Tolerance intervals in statistical software and robustness under model misspecification

  

n = 10

n = 25

n = 50

n = 100

n = 10

n = 25

n = 50

n = 100

  

ρ= 0.9

   

ρ= 0.95

   

α = 0.01

\(\hat {\alpha }\)

0.1824

0.1284

0.0617

0.0140

0.4086

0.3572

0.2632

0.1434

 

\(\hat {\rho }\)

0.9399

0.9479

0.9578

0.9673

0.9495

0.9562

0.9646

0.9725

 

\(\hat {s}\)

0.0493

0.0405

0.0322

0.0244

0.0416

0.0341

0.0271

0.0205

α= 0.05

\(\hat {\alpha }\)

0.2827

0.1771

0.0834

0.0195

0.5065

0.4151

0.2983

0.1658

 

\(\hat {\rho }\)

0.9240

0.9411

0.9543

0.9655

0.9362

0.9505

0.9616

0.9710

 

\(\hat {s}\)

0.0618

0.0456

0.0349

0.0257

0.0523

0.0384

0.0293

0.0216

α= 0.1

\(\hat {\alpha }\)

0.3370

0.2064

0.0966

0.0226

0.5579

0.4420

0.3163

0.1788

 

\(\hat {\rho }\)

0.9149

0.9373

0.9523

0.9645

0.9285

0.9474

0.9600

0.9702

 

\(\hat {s}\)

0.0688

0.0484

0.0363

0.0264

0.0583

0.0408

0.0306

0.0222

α= 0.2

\(\hat {\alpha }\)

0.3962

0.2406

0.1131

0.0276

0.6042

0.4736

0.3404

0.1909

 

\(\hat {\rho }\)

0.9031

0.9326

0.9500

0.9633

0.9186

0.9434

0.9580

0.9692

 

\(\hat {s}\)

0.0776

0.0519

0.0381

0.0273

0.0660

0.0438

0.0321

0.0229

  

ρ= 0.99

   

ρ= 0.995

   

α = 0.01

\(\hat {\alpha }\)

0.8260

0.8059

0.7660

0.7049

0.9058

0.8944

0.8733

0.8399

 

\(\hat {\rho }\)

0.9616

0.9667

0.9730

0.9791

0.9647

0.9694

0.9753

0.9808

 

\(\hat {s}\)

0.0318

0.0260

0.0207

0.0156

0.0293

0.0239

0.0190

0.0143

α= 0.05

\(\hat {\alpha }\)

0.8619

0.8272

0.7823

0.7201

0.9236

0.9064

0.8814

0.8484

 

\(\hat {\rho }\)

0.9514

0.9624

0.9708

0.9780

0.9554

0.9655

0.9732

0.9798

 

\(\hat {s}\)

0.0401

0.0294

0.0224

0.0164

0.0369

0.0270

0.0206

0.0151

α= 0.1

\(\hat {\alpha }\)

0.8781

0.8381

0.7921

0.7278

0.9303

0.9113

0.8864

0.8530

 

\(\hat {\rho }\)

0.9455

0.9599

0.9696

0.9773

0.9500

0.9632

0.9721

0.9792

 

\(\hat {s}\)

0.0448

0.0312

0.0233

0.0169

0.0412

0.0287

0.0214

0.0155

α= 0.2

\(\hat {\alpha }\)

0.8930

0.8499

0.8029

0.7355

0.9395

0.9173

0.8915

0.8572

 

\(\hat {\rho }\)

0.9380

0.9569

0.9681

0.9766

0.9431

0.9605

0.9707

0.9785

 

\(\hat {s}\)

0.0508

0.0335

0.0245

0.0175

0.0468

0.0308

0.0225

0.0160

  1. The bolded values are \({\hat \alpha }\) within \(\pm 2 \sqrt {\alpha (1-\alpha)/M}\) and \({\hat \rho }\) within \(\pm 2 \sqrt {\rho (1-\rho)/M}\)