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Table 18 Performance of parametric tolerance intervals based on the model selection approach for skewed distributions when the true distribution is gamma

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

\(\hat {\alpha }\)

0.0089

0.0106

0.0088

0.0084

0.0094

0.0111

0.0090

0.0090

 

\(\hat {\rho }\)

0.9970

0.9966

0.9968

0.9969

0.9980

0.9978

0.9978

0.9979

 

\(\hat {s}\)

0.0269

0.0277

0.0304

0.0297

0.0221

0.0219

0.0260

0.0248

α= 0.05

\(\hat {\alpha }\)

0.0387

0.0411

0.0397

0.0385

0.0425

0.0440

0.0433

0.0426

 

\(\hat {\rho }\)

0.9868

0.9858

0.9865

0.9867

0.9911

0.9903

0.9909

0.9910

 

\(\hat {s}\)

0.0563

0.0595

0.0581

0.0577

0.0457

0.0488

0.0480

0.0477

α= 0.1

\(\hat {\alpha }\)

0.0731

0.0779

0.0743

0.0752

0.0782

0.0835

0.0807

0.0807

 

\(\hat {\rho }\)

0.9752

0.9737

0.9748

0.9751

0.9831

0.9819

0.9828

0.9830

 

\(\hat {s}\)

0.0767

0.0804

0.0783

0.0777

0.0631

0.0666

0.0647

0.0643

α= 0.2

\(\hat {\alpha }\)

0.1387

0.1471

0.1415

0.1386

0.1518

0.1581

0.1517

0.1490

 

\(\hat {\rho }\)

0.9536

0.9516

0.9532

0.9536

0.9679

0.9662

0.9675

0.9679

 

\(\hat {s}\)

0.1032

0.1073

0.1049

0.1042

0.0861

0.0900

0.0876

0.0871

  

ρ= 0.99

   

ρ= 0.995

   

α = 0.01

\(\hat {\alpha }\)

0.0110

0.0133

0.0116

0.0103

0.0119

0.0142

0.0121

0.0106

 

\(\hat {\rho }\)

0.9989

0.9988

0.9987

0.9988

0.9991

0.9990

0.9989

0.9990

 

\(\hat {s}\)

0.0167

0.0151

0.0205

0.0186

0.0155

0.0134

0.0191

0.0169

α= 0.05

\(\hat {\alpha }\)

0.0501

0.0520

0.0502

0.0506

0.0534

0.0549

0.0530

0.0539

 

\(\hat {\rho }\)

0.9952

0.9946

0.9950

0.9951

0.9960

0.9955

0.9958

0.9959

 

\(\hat {s}\)

0.0330

0.0354

0.0362

0.0356

0.0299

0.0320

0.0333

0.0326

α= 0.1

\(\hat {\alpha }\)

0.0940

0.0977

0.0936

0.0959

0.0994

0.1048

0.0991

0.1003

 

\(\hat {\rho }\)

0.9907

0.9898

0.9905

0.9906

0.9923

0.9915

0.9921

0.9922

 

\(\hat {s}\)

0.0458

0.0491

0.0481

0.0479

0.0414

0.0447

0.0441

0.0437

α= 0.2

\(\hat {\alpha }\)

0.1774

0.1800

0.1755

0.1745

0.1878

0.1894

0.1852

0.1845

 

\(\hat {\rho }\)

0.9821

0.9808

0.9818

0.9819

0.9850

0.9838

0.9847

0.9849

 

\(\hat {s}\)

0.0636

0.0674

0.0653

0.0651

0.0578

0.0615

0.0596

0.0595

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