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Table 20 Performance of parametric tolerance intervals based on the model selection approach for skewed distributions when the true distribution is lognormal

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

\(\hat {\alpha }\)

0.0124

0.0096

0.0128

0.0112

0.0141

0.0115

0.0142

0.0140

 

\(\hat {\rho }\)

0.9961

0.9963

0.9958

0.9957

0.9973

0.9974

0.9972

0.9970

 

\(\hat {s}\)

0.0277

0.0301

0.0291

0.0335

0.0222

0.0253

0.0234

0.0287

α= 0.05

\(\hat {\alpha }\)

0.0483

0.0425

0.0498

0.0485

0.0557

0.0524

0.0607

0.0549

 

\(\hat {\rho }\)

0.9837

0.9844

0.9824

0.9830

0.9885

0.9892

0.9877

0.9880

 

\(\hat {s}\)

0.0599

0.0587

0.0622

0.0634

0.0493

0.0488

0.0512

0.0531

α= 0.1

\(\hat {\alpha }\)

0.0847

0.0834

0.0934

0.0853

0.0977

0.0973

0.1101

0.1023

 

\(\hat {\rho }\)

0.9710

0.9715

0.9685

0.9699

0.9792

0.9799

0.9775

0.9784

 

\(\hat {s}\)

0.0804

0.0786

0.0838

0.0838

0.0672

0.0656

0.0698

0.0706

α= 0.2

\(\hat {\alpha }\)

0.1519

0.1574

0.1652

0.1543

0.1768

0.1827

0.1901

0.1809

 

\(\hat {\rho }\)

0.9490

0.9489

0.9452

0.9474

0.9630

0.9633

0.9600

0.9617

 

\(\hat {s}\)

0.1059

0.1044

0.1108

0.1094

0.0898

0.0879

0.0937

0.0933

  

ρ= 0.99

   

ρ= 0.995

   

α = 0.01

\(\hat {\alpha }\)

0.0183

0.0166

0.0199

0.0180

0.0204

0.0182

0.0217

0.0199

 

\(\hat {\rho }\)

0.9986

0.9985

0.9985

0.9982

0.9988

0.9987

0.9988

0.9984

 

\(\hat {s}\)

0.0158

0.0194

0.0166

0.0231

0.0142

0.0178

0.0149

0.0216

α= 0.05

\(\hat {\alpha }\)

0.0738

0.0715

0.0832

0.0743

0.0820

0.0798

0.0901

0.0810

 

\(\hat {\rho }\)

0.9933

0.9938

0.9930

0.9930

0.9944

0.9948

0.9941

0.9940

 

\(\hat {s}\)

0.0359

0.0367

0.0376

0.0407

0.0325

0.0337

0.0342

0.0376

α= 0.1

\(\hat {\alpha }\)

0.1300

0.1326

0.1446

0.1360

0.1429

0.1469

0.1565

0.1507

 

\(\hat {\rho }\)

0.9877

0.9884

0.9868

0.9872

0.9895

0.9902

0.9888

0.9891

 

\(\hat {s}\)

0.0501

0.0493

0.0521

0.0539

0.0456

0.0452

0.0476

0.0497

α= 0.2

\(\hat {\alpha }\)

0.2334

0.2409

0.2451

0.2403

0.2520

0.2622

0.2669

0.2620

 

\(\hat {\rho }\)

0.9775

0.9782

0.9757

0.9767

0.9807

0.9815

0.9792

0.9800

 

\(\hat {s}\)

0.0686

0.0667

0.0711

0.0720

0.0629

0.0612

0.0653

0.0664

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