Table 3: Estimated MSE when p = 3, n = 100.

k

d

sigma

rho

OLS

RIDGE

LIU

K-L

MRT

DK

LDK

0.3

0.2

1

0.8

0.0622

0.0611

0.0594

0.06

0.0609

0.0596

0.057

0.9

0.1131

0.1093

0.1036

0.1056

0.1086

0.1042

0.0955

0.95

0.2163

0.2023

0.1828

0.1889

0.1997

0.1839

0.1556

0.99

1.0445

0.7667

0.5345

0.5329

0.7248

0.4649

0.2416

3

0.8

0.5596

0.5499

0.5345

0.5403

0.548

0.5366

0.5127

0.9

1.0178

0.9839

0.9322

0.9506

0.9774

0.9378

0.8593

0.95

1.9464

1.8211

1.6447

1.7002

1.7976

1.6549

1.4001

0.99

9.4002

6.9

4.8107

4.7956

6.5235

4.1841

2.1739

5

0.8

1.5543

1.5275

1.4848

1.5009

1.5222

1.4905

1.424

0.9

2.8273

2.7332

2.5896

2.6407

2.7149

2.6049

2.3869

0.95

5.4066

5.0586

4.5687

4.7227

4.9932

4.597

3.8892

0.99

26.112

19.167

13.363

13.321

18.121

11.623

6.0386

10

0.8

6.2173

6.11

5.9393

6.0036

6.0889

5.9618

5.6961

0.9

11.309

10.933

10.358

10.563

10.86

10.42

9.5474

0.95

21.626

20.235

18.275

18.891

19.973

18.388

15.557

0.99

104.45

76.667

53.452

53.285

72.483

46.49

24.154

 

0.3

0.5

1

0.8

0.0622

0.0611

0.0604

0.06

0.0606

0.059

0.0574

0.9

0.1131

0.1093

0.1071

0.1056

0.1075

0.1021

0.0967

0.95

0.2163

0.2023

0.195

0.1889

0.1959

0.1766

0.1594

0.99

1.0445

0.7667

0.7055

0.5329

0.6683

0.3779

0.258

3

0.8

0.5596

0.5499

0.5438

0.5403

0.5452

0.531

0.5162

0.9

1.0178

0.9839

0.9639

0.9506

0.9677

0.9188

0.8704

0.95

1.9464

1.8211

1.7548

1.7002

1.7631

1.5894

1.4342

0.99

9.4002

6.9

6.3497

4.7956

6.015

3.4012

2.322

5

0.8

1.5543

1.5275

1.5107

1.5009

1.5144

1.4749

1.4337

0.9

2.8273

2.7332

2.6774

2.6407

2.6879

2.5523

2.4177

0.95

5.4066

5.0586

4.8743

4.7227

4.8976

4.4148

3.9838

0.99

26.112

19.167

17.638

13.321

16.708

9.4477

6.4499

10

0.8

6.2173

6.11

6.0427

6.0036

6.0574

5.8998

5.7348

0.9

11.309

10.933

10.71

10.563

10.752

10.209

9.6709

0.95

21.626

20.235

19.497

18.891

19.59

17.659

15.935

0.99

104.45

76.667

70.552

53.285

66.834

37.791

25.799

 

0.3

0.8

1

0.8

0.0622

0.0611

0.0615

0.06

0.0603

0.0584

0.0578

0.9

0.1131

0.1093

0.1107

0.1056

0.1065

0.1

0.0979

0.95

0.2163

0.2023

0.2076

0.1889

0.1922

0.1696

0.1629

0.99

1.0445

0.7667

0.9008

0.5329

0.6183

0.306

0.2651

3

0.8

0.5596

0.5499

0.5532

0.5403

0.5424

0.5255

0.5196

0.9

1.0178

0.9839

0.9961

0.9506

0.9581

0.9003

0.8812

0.95

1.9464

1.8211

1.8685

1.7002

1.7297

1.5264

1.466

0.99

9.4002

6.9

8.1072

4.7956

5.5649

2.7536

2.3861

5

0.8

1.5543

1.5275

1.5368

1.5009

1.5066

1.4596

1.4432

0.9

2.8273

2.7332

2.7668

2.6407

2.6614

2.5007

2.4476

0.95

5.4066

5.0586

5.1902

4.7227

4.8048

4.24

4.072

0.99

26.112

19.167

22.52

13.321

15.458

7.649

6.6279

10

0.8

6.2173

6.11

6.1472

6.0036

6.0262

5.8384

5.7728

0.9

11.309

10.933

11.067

10.563

10.645

10.003

9.7903

0.95

21.626

20.235

20.761

18.891

19.219

16.96

16.288

0.99

104.45

76.667

90.08

53.285

61.832

30.596

26.511

 

0.6

0.2

1

0.8

0.0622

0.0601

0.0594

0.058

0.0597

0.0572

0.0547

0.9

0.1131

0.1058

0.1036

0.0987

0.1044

0.0961

0.0881

0.95

0.2163

0.1898

0.1828

0.1651

0.1851

0.1565

0.1326

0.99

1.0445

0.5881

0.5345

0.2651

0.5341

0.1972

0.105

3

0.8

0.5596

0.5405

0.5345

0.5218

0.5368

0.5146

0.4918

0.9

1.0178

0.9518

0.9322

0.8881

0.9394

0.8643

0.7924

0.95

1.9464

1.708

1.6447

1.4859

1.6658

1.408

1.1928

0.99

9.4002

5.2929

4.8107

2.3853

4.8065

1.775

0.944

5

0.8

1.5543

1.5014

1.4848

1.4495

1.4912

1.4294

1.366

0.9

2.8273

2.6439

2.5896

2.4669

2.6094

2.4008

2.2009

0.95

5.4066

4.7444

4.5687

4.1273

4.6272

3.911

3.3131

0.99

26.112

14.702

13.363

6.6258

13.351

4.9303

2.6219

10

0.8

6.2173

6.0056

5.9393

5.7978

5.9646

5.7176

5.4637

0.9

11.309

10.575

10.358

9.8677

10.438

9.6032

8.8032

0.95

21.626

18.978

18.275

16.509

18.509

15.644

13.252

0.99

104.45

58.81

53.452

26.503

53.405

19.721

10.487

 

0.6

0.5

1

0.8

0.0622

0.0601

0.0604

0.058

0.0591

0.0561

0.0545

0.9

0.1131

0.1058

0.1071

0.0987

0.1024

0.0923

0.0875

0.95

0.2163

0.1898

0.195

0.1651

0.1784

0.1444

0.1304

0.99

1.0445

0.5881

0.7055

0.2651

0.4662

0.1233

0.0859

3

0.8

0.5596

0.5405

0.5438

0.5218

0.5314

0.504

0.49

0.9

1.0178

0.9518

0.9639

0.8881

0.9213

0.8299

0.7864

0.95

1.9464

1.708

1.7548

1.4859

1.6055

1.2988

1.1732

0.99

9.4002

5.2929

6.3497

2.3853

4.1956

1.1091

0.7724

5

0.8

1.5543

1.5014

1.5107

1.4495

1.476

1.3999

1.361

0.9

2.8273

2.6439

2.6774

2.4669

2.5591

2.3051

2.1844

0.95

5.4066

4.7444

4.8743

4.1273

4.4597

3.6077

3.2587

0.99

26.112

14.702

17.638

6.6258

11.654

3.0808

2.1454

10

0.8

6.2173

6.0056

6.0427

5.7978

5.9041

5.5996

5.4437

0.9

11.309

10.575

10.71

9.8677

10.236

9.2202

8.7372

0.95

21.626

18.978

19.497

16.509

17.839

14.431

13.034

0.99

104.45

58.81

70.552

26.503

46.617

12.323

8.5812

 

0.6

0.8

1

0.8

0.0622

0.0601

0.0615

0.058

0.0585

0.0549

0.0543

0.9

0.1131

0.1058

0.1107

0.0987

0.1004

0.0886

0.0868

0.95

0.2163

0.1898

0.2076

0.1651

0.1721

0.1332

0.128

0.99

1.0445

0.5881

0.9008

0.2651

0.4107

0.0739

0.0647

3

0.8

0.5596

0.5405

0.5532

0.5218

0.526

0.4936

0.4881

0.9

1.0178

0.9518

0.9961

0.8881

0.9037

0.7968

0.7801

0.95

1.9464

1.708

1.8685

1.4859

1.5485

1.1981

1.1512

0.99

9.4002

5.2929

8.1072

2.3853

3.6959

0.6642

0.582

5

0.8

1.5543

1.5014

1.5368

1.4495

1.4611

1.3711

1.3558

0.9

2.8273

2.6439

2.7668

2.4669

2.5103

2.2133

2.1667

0.95

5.4066

4.7444

5.1902

4.1273

4.3015

3.3278

3.1975

0.99

26.112

14.702

22.52

6.6258

10.266

1.845

1.6166

10

0.8

6.2173

6.0056

6.1472

5.7978

5.8445

5.4841

5.4229

0.9

11.309

10.575

11.067

9.8677

10.041

8.853

8.6664

0.95

21.626

18.978

20.761

16.509

17.206

13.311

12.79

0.99

104.45

58.81

90.08

26.503

41.065

7.3796

6.4659

 

0.7

0.2

1

0.8

0.0622

0.0597

0.0594

0.0573

0.0593

0.0564

0.054

0.9

0.1131

0.1046

0.1036

0.0965

0.103

0.0935

0.0858

0.95

0.2163

0.1859

0.1828

0.1579

0.1806

0.1483

0.1257

0.99

1.0445

0.5425

0.5345

0.2074

0.4873

0.1448

0.078

3

0.8

0.5596

0.5374

0.5345

0.5158

0.5332

0.5075

0.4851

0.9

1.0178

0.9415

0.9322

0.8682

0.9273

0.8412

0.7713

0.95

1.9464

1.6727

1.6447

1.4207

1.6252

1.3342

1.1308

0.99

9.4002

4.8827

4.8107

1.8662

4.3853

1.3027

0.7009

5

0.8

1.5543

1.4929

1.4848

1.4327

1.481

1.4097

1.3472

0.9

2.8273

2.6151

2.5896

2.4117

2.5757

2.3365

2.1423

0.95

5.4066

4.6464

4.5687

3.9463

4.5145

3.7061

3.1409

0.99

26.112

13.563

13.363

5.1838

12.181

3.6186

1.9468

10

0.8

6.2173

5.9714

5.9393

5.7309

5.9241

5.6386

5.3885

0.9

11.309

10.461

10.358

9.6467

10.303

9.346

8.5688

0.95

21.626

18.586

18.275

15.785

18.058

14.824

12.563

0.99

104.45

54.252

53.452

20.735

48.726

14.474

7.7867

 

0.7

0.5

1

0.8

0.0622

0.0597

0.0604

0.0573

0.0586

0.0551

0.0536

0.9

0.1131

0.1046

0.1071

0.0965

0.1008

0.0892

0.0846

0.95

0.2163

0.1859

0.195

0.1579

0.1731

0.135

0.122

0.99

1.0445

0.5425

0.7055

0.2074

0.4192

0.0807

0.0569

3

0.8

0.5596

0.5374

0.5438

0.5158

0.5269

0.4954

0.4816

0.9

1.0178

0.9415

0.9639

0.8682

0.9066

0.8022

0.7604

0.95

1.9464

1.6727

1.7548

1.4207

1.5578

1.2143

1.0972

0.99

9.4002

4.8827

6.3497

1.8662

3.7726

0.726

0.5117

5

0.8

1.5543

1.4929

1.5107

1.4327

1.4636

1.3758

1.3376

0.9

2.8273

2.6151

2.6774

2.4117

2.5184

2.2283

2.1119

0.95

5.4066

4.6464

4.8743

3.9463

4.3272

3.3729

3.0477

0.99

26.112

13.563

17.638

5.1838

10.48

2.0165

1.4211

10

0.8

6.2173

5.9714

6.0427

5.7309

5.8543

5.5032

5.3502

0.9

11.309

10.461

10.71

9.6467

10.073

8.9131

8.4472

0.95

21.626

18.586

19.497

15.785

17.309

13.491

12.19

0.99

104.45

54.252

70.552

20.735

41.918

8.0656

5.6838

 

0.7

0.8

1

0.8

0.0622

0.0597

0.0615

0.0573

0.0579

0.0538

0.0533

0.9

0.1131

0.1046

0.1107

0.0965

0.0985

0.0851

0.0834

0.95

0.2163

0.1859

0.2076

0.1579

0.1661

0.1229

0.1181

0.99

1.0445

0.5425

0.9008

0.2074

0.3647

0.0421

0.0372

3

0.8

0.5596

0.5374

0.5532

0.5158

0.5207

0.4835

0.4781

0.9

1.0178

0.9415

0.9961

0.8682

0.8867

0.7652

0.7491

0.95

1.9464

1.6727

1.8685

1.4207

1.4947

1.105

1.062

0.99

9.4002

4.8827

8.1072

1.8662

3.2818

0.3778

0.3338

5

0.8

1.5543

1.4929

1.5368

1.4327

1.4465

1.3429

1.3279

0.9

2.8273

2.6151

2.7668

2.4117

2.463

2.1253

2.0806

0.95

5.4066

4.6464

5.1902

3.9463

4.1518

3.0694

2.9497

0.99

26.112

13.563

22.52

5.1838

9.1161

1.0491

0.9269

10

0.8

6.2173

5.9714

6.1472

5.7309

5.7859

5.3712

5.3114

0.9

11.309

10.461

11.067

9.6467

9.8519

8.5008

8.3221

0.95

21.626

18.586

20.761

15.785

16.607

12.277

11.798

0.99

104.45

54.252

90.08

20.735

36.464

4.1961

3.7071

 

0.9

0.2

1

0.8

0.0622

0.0591

0.0594

0.0561

0.0585

0.0549

0.0526

0.9

0.1131

0.1024

0.1036

0.0923

0.1004

0.0886

0.0813

0.95

0.2163

0.1784

0.1828

0.1444

0.1721

0.1332

0.1131

0.99

1.0445

0.4662

0.5345

0.1233

0.4107

0.0739

0.0412

3

0.8

0.5596

0.5314

0.5345

0.504

0.526

0.4936

0.4719

0.9

1.0178

0.9213

0.9322

0.8299

0.9037

0.7968

0.7309

0.95

1.9464

1.6055

1.6447

1.2988

1.5485

1.1981

1.0164

0.99

9.4002

4.1956

4.8107

1.1091

3.6959

0.6642

0.37

5

0.8

1.5543

1.476

1.4848

1.3999

1.4611

1.3711

1.3105

0.9

2.8273

2.5591

2.5896

2.3051

2.5103

2.2133

2.0299

0.95

5.4066

4.4597

4.5687

3.6077

4.3015

3.3278

2.8229

0.99

26.112

11.654

13.363

3.0808

10.266

1.845

1.0276

10

0.8

6.2173

5.9041

5.9393

5.5996

5.8445

5.4841

5.2415

0.9

11.309

10.236

10.358

9.2202

10.041

8.853

8.1192

0.95

21.626

17.839

18.275

14.431

17.206

13.311

11.291

0.99

104.45

46.617

53.452

12.323

41.065

7.3796

4.1099

 
 

0.5

1

0.8

0.0622

0.0591

0.0604

0.0561

0.0576

0.0533

0.0519

0.9

0.1131

0.1024

0.1071

0.0923

0.0976

0.0834

0.0791

0.95

0.2163

0.1784

0.195

0.1444

0.1632

0.118

0.1068

0.99

1.0445

0.4662

0.7055

0.1233

0.3446

0.0312

0.023

3

0.8

0.5596

0.5314

0.5438

0.504

0.5181

0.4785

0.4653

0.9

1.0178

0.9213

0.9639

0.8299

0.8784

0.7498

0.7109

0.95

1.9464

1.6055

1.7548

1.2988

1.4688

1.0612

0.9597

0.99

9.4002

4.1956

6.3497

1.1091

3.1008

0.2798

0.2059

5

0.8

1.5543

1.476

1.5107

1.3999

1.4392

1.329

1.2922

0.9

2.8273

2.5591

2.6774

2.3051

2.4399

2.0826

1.9743

0.95

5.4066

4.4597

4.8743

3.6077

4.08

2.9476

2.6654

0.99

26.112

11.654

17.638

3.0808

8.6133

0.777

0.5716

10

0.8

6.2173

5.9041

6.0427

5.5996

5.7569

5.3157

5.1685

0.9

11.309

10.236

10.71

9.2202

9.7593

8.3302

7.8967

0.95

21.626

17.839

19.497

14.431

16.32

11.79

10.661

0.99

104.45

46.617

70.552

12.323

34.453

3.1074

2.2859

 

0.9

0.8

1

0.8

0.0622

0.0591

0.0615

0.0561

0.0568

0.0518

0.0512

0.9

0.1131

0.1024

0.1107

0.0923

0.0949

0.0786

0.077

0.95

0.2163

0.1784

0.2076

0.1444

0.1551

0.1046

0.1005

0.99

1.0445

0.4662

0.9008

0.1233

0.2934

0.0133

0.0122

3

0.8

0.5596

0.5314

0.5532

0.504

0.5105

0.4639

0.4588

0.9

1.0178

0.9213

0.9961

0.8299

0.8541

0.7057

0.691

0.95

1.9464

1.6055

1.8685

1.2988

1.3954

0.9397

0.9034

0.99

9.4002

4.1956

8.1072

1.1091

2.6407

0.1189

0.1082

5

0.8

1.5543

1.476

1.5368

1.3999

1.4179

1.2883

1.2741

0.9

2.8273

2.5591

2.7668

2.3051

2.3725

1.9599

1.9189

0.95

5.4066

4.4597

5.1902

3.6077

3.8759

2.6099

2.5091

0.99

26.112

11.654

22.52

3.0808

7.3351

0.3301

0.3001

10

0.8

6.2173

5.9041

6.1472

5.5996

5.6715

5.1528

5.0956

0.9

11.309

10.236

11.067

9.2202

9.4898

7.839

7.6751

0.95

21.626

17.839

20.761

14.431

15.503

10.439

10.036

0.99

104.45

46.617

90.08

12.323

29.34

1.3197

1.1998

Note: For each row, the smallest MSE value is bolded.