Table 2: Estimated MSE when p = 3, n = 50.

k

d

sigma

rho

OLS

RIDGE

LIU

K-L

MRT

DK

LDK

0.3

0.2

1

0.8

0.1363

0.1311

0.1234

0.1261

0.1301

0.1241

0.1125

0.9

0.2462

0.2288

0.2046

0.212

0.2255

0.2057

0.1713

0.95

0.4682

0.4064

0.3333

0.3492

0.3956

0.3293

0.2355

0.99

2.2417

1.2173

0.7223

0.5101

1.0998

0.3689

0.1264

3

0.8

0.9

1.227

2.2158

1.1802

2.0587

1.1098

1.8412

1.1344

1.9076

1.1712

2.0293

1.1168

1.8514

1.0107

1.5407

 

0.95

4.2136

3.6579

2.9999

3.1428

3.5604

2.964

2.1191

0.99

20.175

10.956

6.5009

4.5913

9.8985

3.3199

1.1375

5

0.8

3.4082

3.2782

3.0825

3.1509

3.2532

3.1019

2.8071

0.9

6.1551

5.7185

5.1142

5.2987

5.637

5.1426

4.2792

0.95

11.704

10.161

8.333

8.73

9.8901

8.2333

5.8864

0.99

56.042

30.434

18.058

12.754

27.496

9.2219

3.1597

10

0.8

13.633

13.113

12.329

12.603

13.012

12.407

11.227

0.9

24.62

22.874

20.457

21.194

22.548

20.57

17.116

0.95

46.818

40.644

33.332

34.92

39.56

32.933

23.546

0.99

224.17

121.73

72.233

51.015

109.98

36.888

12.639

 

0.3

0.5

1

0.8

0.1363

0.1311

0.1281

0.1261

0.1287

0.1213

0.1141

0.9

0.2462

0.2288

0.2197

0.212

0.2207

0.1968

0.1758

0.95

0.4682

0.4064

0.3811

0.3492

0.3802

0.3016

0.2463

0.99

2.2417

1.2173

1.1925

0.5101

0.9535

0.2188

0.1217

3

0.8

1.227

1.1802

1.153

1.1344

1.1578

1.0909

1.0256

0.9

2.2158

2.0587

1.9774

1.9076

1.9865

1.7703

1.5816

0.95

4.2136

3.6579

3.4302

3.1428

3.4215

2.7145

2.2167

0.99

20.175

10.956

10.733

4.5913

8.5816

1.9692

1.0948

5

0.8

3.4082

3.2782

3.2026

3.1509

3.2161

3.03

2.8484

0.9

6.1551

5.7185

5.4928

5.2987

5.5181

4.9173

4.3931

0.95

11.704

10.161

9.5284

8.73

9.5043

7.5403

6.1574

0.99

56.042

30.434

29.813

12.754

23.838

5.4701

3.041

10

0.8

13.633

13.113

12.81

12.603

12.864

12.119

11.393

0.9

24.62

22.874

21.971

21.194

22.072

19.669

17.572

0.95

46.818

40.644

38.114

34.92

38.017

30.161

24.63

0.99

224.17

121.73

119.25

51.015

95.351

21.881

12.164

 

0.3

0.8

1

0.8

0.1363

0.1311

0.133

0.1261

0.1272

0.1185

0.1157

0.9

0.2462

0.2288

0.2354

0.212

0.2162

0.1882

0.18

0.95

0.4682

0.4064

0.4323

0.3492

0.3657

0.2762

0.2554

0.99

2.2417

1.2173

1.7822

0.5101

0.8349

0.1225

0.0996

3

0.8

1.227

1.1802

1.1971

1.1344

1.1448

1.0656

1.0399

0.9

2.2158

2.0587

2.1188

1.9076

1.9451

1.6928

1.6195

0.95

4.2136

3.6579

3.8903

3.1428

3.2909

2.4857

2.2982

0.99

20.175

10.956

16.04

4.5913

7.5144

1.1027

0.896

5

0.8

3.4082

3.2782

3.3251

3.1509

3.1798

2.9597

2.8882

0.9

6.1551

5.7185

5.8855

5.2987

5.4031

4.702

4.4984

0.95

11.704

10.161

10.806

8.73

9.1413

6.9046

6.3839

0.99

56.042

30.434

44.554

12.754

20.873

3.063

2.489

10

0.8

13.633

13.113

13.3

12.603

12.719

11.838

11.552

0.9

24.62

22.874

23.542

21.194

21.612

18.808

17.993

0.95

46.818

40.644

43.226

34.92

36.565

27.619

25.536

0.99

224.17

121.73

178.22

51.015

83.494

12.252

9.9562

 

0.6

0.2

1

0.8

0.1363

0.1263

0.1234

0.1167

0.1244

0.1131

0.1026

0.9

0.2462

0.2132

0.2046

0.1827

0.2074

0.1721

0.1436

0.95

0.4682

0.3564

0.3333

0.2604

0.3391

0.2314

0.1664

0.99

2.2417

0.7679

0.7223

0.0801

0.6563

0.0327

0.0153

3

0.8

1.227

1.1362

1.1098

1.0491

1.1193

1.0169

0.921

0.9

2.2158

1.9183

1.8412

1.6431

1.8663

1.548

1.2903

0.95

4.2136

3.208

2.9999

2.3436

3.0516

2.0827

1.4966

0.99

20.175

6.9114

6.5009

0.721

5.907

0.2939

0.1371

5

0.8

3.4082

3.1559

3.0825

2.9139

3.109

2.8244

2.5574

0.9

6.1551

5.3284

5.1142

4.5638

5.1839

4.2996

3.5836

0.95

11.704

8.9111

8.333

6.5101

8.4768

5.7852

4.1572

0.99

56.042

19.198

18.058

2.0029

16.408

0.8163

0.3806

10

0.8

13.633

12.623

12.329

11.655

12.436

11.296

10.228

0.9

24.62

21.313

20.457

18.255

20.735

17.198

14.333

0.95

46.818

35.645

33.332

26.041

33.907

23.141

16.629

0.99

224.17

76.794

72.233

8.0117

65.634

3.2652

1.5225

 

0.6

0.5

1

0.8

0.1363

0.1263

0.1281

0.1167

0.1217

0.1081

0.1018

0.9

0.2462

0.2132

0.2197

0.1827

0.1992

0.1575

0.141

0.95

0.4682

0.3564

0.3811

0.2604

0.3154

0.1937

0.1589

0.99

2.2417

0.7679

1.1925

0.0801

0.5301

0.0227

0.0144

3

0.8

1.227

1.1362

1.153

1.0491

1.0947

0.9706

0.913

0.9

2.2158

1.9183

1.9774

1.6431

1.7923

1.4156

1.2664

0.95

4.2136

3.208

3.4302

2.3436

2.8383

1.7424

1.4286

0.99

20.175

6.9114

10.733

0.721

4.7705

0.2033

0.1281

5

0.8

3.4082

3.1559

3.2026

2.9139

3.0407

2.6954

2.5351

0.9

6.1551

5.3284

5.4928

4.5638

4.9784

3.9317

3.517

0.95

11.704

8.9111

9.5284

6.5101

7.8841

4.84

3.9681

0.99

56.042

19.198

29.813

2.0029

13.251

0.5647

0.3558

10

0.8

13.633

12.623

12.81

11.655

12.162

10.78

10.138

0.9

24.62

21.313

21.971

18.255

19.913

15.726

14.066

0.95

46.818

35.645

38.114

26.041

31.537

19.36

15.873

0.99

224.17

76.794

119.25

8.0117

53.006

2.2591

1.4231

 

0.6

0.8

1

0.8

0.1363

0.1263

0.133

0.1167

0.1191

0.1032

0.1008

0.9

0.2462

0.2132

0.2354

0.1827

0.1915

0.1441

0.138

0.95

0.4682

0.3564

0.4323

0.2604

0.2941

0.1617

0.1499

0.99

2.2417

0.7679

1.7822

0.0801

0.4374

0.0543

0.0437

3

0.8

1.227

1.1362

1.1971

1.0491

1.071

0.9264

0.9043

0.9

2.2158

1.9183

2.1188

1.6431

1.7228

1.2944

1.2392

0.95

4.2136

3.208

3.8903

2.3436

2.6472

1.4547

1.3476

0.99

20.175

6.9114

16.04

0.721

3.9366

0.4881

0.3923

5

0.8

3.4082

3.1559

3.3251

2.9139

2.9747

2.5725

2.5109

0.9

6.1551

5.3284

5.8855

4.5638

4.7853

3.595

3.4414

0.95

11.704

8.9111

10.806

6.5101

7.3532

4.0406

3.7432

0.99

56.042

19.198

44.554

2.0029

10.935

1.3556

1.0897

10

0.8

13.633

12.623

13.3

11.655

11.898

10.288

10.042

0.9

24.62

21.313

23.542

18.255

19.141

14.379

13.764

0.95

46.818

35.645

43.226

26.041

29.413

16.163

14.973

0.99

224.17

76.794

178.22

8.0117

43.741

5.4227

4.3589

 

0.7

0.2

1

0.8

0.1363

0.1247

0.1234

0.1137

0.1226

0.1097

0.0996

0.9

0.2462

0.2083

0.2046

0.1738

0.2019

0.1622

0.1355

0.95

0.4682

0.3419

0.3333

0.2361

0.323

0.2056

0.1481

0.99

2.2417

0.6731

0.7223

0.0378

0.5677

0.02

0.0103

3

0.8

1.227

1.1221

1.1098

1.0222

1.1028

0.9858

0.893

0.9

2.2158

1.8748

1.8412

1.5634

1.8164

1.4584

1.2164

0.95

4.2136

3.0769

2.9999

2.1241

2.9068

1.8495

1.3317

0.99

20.175

6.0583

6.5009

0.3396

5.1093

0.1793

0.0912

5

0.8

3.4082

3.1167

3.0825

2.8391

3.0632

2.7377

2.4795

0.9

6.1551

5.2076

5.1142

4.3425

5.0455

4.0507

3.3781

0.95

11.704

8.5469

8.333

5.9004

8.0744

5.1375

3.6991

0.99

56.042

16.829

18.058

0.9434

14.192

0.4981

0.2532

10

0.8

13.633

12.467

12.329

11.355

12.252

10.949

9.9158

0.9

24.62

20.83

20.457

17.369

20.182

16.202

13.511

0.95

46.818

34.188

33.332

23.602

32.298

20.55

14.796

0.99

224.17

67.315

72.233

3.7737

56.77

1.9927

1.0128

 

0.7

0.5

1

0.8

0.1363

0.1247

0.1281

0.1137

0.1195

0.104

0.098

0.9

0.2462

0.2083

0.2197

0.1738

0.1927

0.1462

0.131

0.95

0.4682

0.3419

0.3811

0.2361

0.2975

0.1667

0.137

0.99

2.2417

0.6731

1.1925

0.0378

0.4511

0.0472

0.0266

3

0.8

1.227

1.1221

1.153

1.0222

1.0749

0.9336

0.8784

0.9

2.2158

1.8748

1.9774

1.5634

1.7341

1.3139

1.176

0.95

4.2136

3.0769

3.4302

2.1241

2.6776

1.4993

1.2313

0.99

20.175

6.0583

10.733

0.3396

4.0594

0.4235

0.2377

5

0.8

3.1167

3.2026

2.8391

3.2573

3.2607

2.4388

2.3475

0.9

6.1551

5.2076

5.4928

4.3425

4.8167

3.6491

3.2657

0.95

11.704

8.5469

9.5284

5.9004

7.4378

4.1648

3.4201

0.99

56.042

16.829

29.813

0.9434

11.276

1.1763

0.6601

10

0.8

13.633

12.467

12.81

11.355

11.942

10.369

9.7529

0.9

24.62

20.83

21.971

17.369

19.266

14.595

13.061

0.95

46.818

34.188

38.114

23.602

29.751

16.659

13.68

0.99

224.17

67.315

119.25

3.7737

45.104

4.7054

2.6404

 

0.7

0.8

1

0.8

0.1363

0.1247

0.133

0.1137

0.1166

0.0986

0.0964

0.9

0.2462

0.2083

0.2354

0.1738

0.1843

0.1318

0.1263

0.95

0.4682

0.3419

0.4323

0.2361

0.2751

0.1347

0.125

0.99

2.2417

0.6731

1.7822

0.0378

0.3674

0.1077

0.0857

3

0.8

1.227

1.1221

1.1971

1.0222

1.0481

0.8843

0.8633

0.9

2.2158

1.8748

2.1188

1.5634

1.6574

1.1835

1.1333

0.95

4.2136

3.0769

3.8903

2.1241

2.4752

1.2112

1.123

0.99

20.175

6.0583

16.04

0.3396

3.3064

0.9676

0.7701

5

0.8

3.4082

3.1167

3.3251

2.8391

2.911

2.4554

2.3968

0.9

6.1551

5.2076

5.8855

4.3425

4.6037

3.2868

3.1471

0.95

11.704

8.5469

10.806

5.9004

6.8757

3.3642

3.1191

0.99

56.042

16.829

44.554

0.9434

9.1844

2.6876

2.1388

10

0.8

13.633

12.467

13.3

11.355

11.643

9.8194

9.5847

0.9

24.62

20.83

23.542

17.369

18.414

13.145

12.586

0.95

46.818

34.188

43.226

23.602

27.503

13.457

12.476

0.99

224.17

67.315

178.22

3.7737

36.738

10.751

8.5552

 

0.9

0.2

1

0.8

0.1363

0.1217

0.1234

0.1081

0.1191

0.1032

0.0938

0.9

0.2462

0.1992

0.2046

0.1575

0.1915

0.1441

0.1205

0.95

0.4682

0.3154

0.3333

0.1937

0.2941

0.1617

0.1171

0.99

2.2417

0.5301

0.7223

0.0227

0.4374

0.0543

0.0197

3

0.8

1.227

1.0947

1.1098

0.9706

1.071

0.9264

0.8396

0.9

2.2158

1.7923

1.8412

1.4156

1.7228

1.2944

1.081

0.95

4.2136

2.8383

2.9999

1.7424

2.6472

1.4547

1.0522

0.99

20.175

4.7705

6.5009

0.2033

3.9366

0.4881

0.1753

5

0.8

3.4082

3.0407

3.0825

2.6954

2.9747

2.5725

2.3309

0.9

6.1551

4.9784

5.1142

3.9317

4.7853

3.595

3.0019

0.95

11.704

7.8841

8.333

4.84

7.3532

4.0406

2.9224

0.99

56.042

13.251

18.058

0.5647

10.935

1.3556

0.4867

10

0.8

13.633

12.162

12.329

10.78

11.898

10.288

9.3209

0.9

24.62

19.913

20.457

15.726

19.141

14.379

12.005

0.95

46.818

31.537

33.332

19.36

29.413

16.163

11.689

0.99

224.17

53.006

72.233

2.2591

43.741

5.4227

1.9465

 
 

0.5

1

0.8

0.1363

0.1217

0.1281

0.1081

0.1153

0.0964

0.091

0.9

0.2462

0.1992

0.2197

0.1575

0.1808

0.1261

0.1131

0.95

0.4682

0.3154

0.3811

0.1937

0.2662

0.1228

0.1014

0.99

2.2417

0.5301

1.1925

0.0227

0.3386

0.1388

0.0741

3

0.8

1.227

1.0947

1.153

0.9706

1.0369

0.864

0.8132

0.9

2.2158

1.7923

1.9774

1.4156

1.6262

1.1316

1.0139

0.95

4.2136

2.8383

3.4302

1.7424

2.3957

1.1038

0.9101

0.99

20.175

4.7705

10.733

0.2033

3.0474

1.2476

0.6643

5

0.8

3.4082

3.0407

3.2026

2.6954

2.88

2.3989

2.2574

0.9

6.1551

4.9784

5.4928

3.9317

4.5169

3.1425

2.8152

0.95

11.704

7.8841

9.5284

4.84

6.6547

3.066

2.5276

0.99

56.042

13.251

29.813

0.5647

8.4651

3.4652

1.8448

10

0.8

13.633

12.162

12.81

10.78

11.519

9.5932

9.0265

0.9

24.62

19.913

21.971

15.726

18.067

12.568

11.258

0.95

46.818

31.537

38.114

19.36

26.619

12.264

10.11

0.99

224.17

53.006

119.25

2.2591

33.861

13.861

7.3791

 

0.9

0.8

1

0.8

0.1363

0.1217

0.133

0.1081

0.1118

0.0902

0.0881

0.9

0.2462

0.1992

0.2354

0.1575

0.171

0.1104

0.1058

0.95

0.4682

0.3154

0.4323

0.1937

0.2422

0.0926

0.0861

0.99

2.2417

0.5301

1.7822

0.0227

0.2703

0.2401

0.1905

3

0.8

1.227

1.0947

1.1971

0.9706

1.0046

0.806

0.787

0.9

2.2158

1.7923

2.1188

1.4156

1.5378

0.9888

0.9473

0.95

4.2136

2.8383

3.8903

1.7424

2.1794

0.8309

0.772

0.99

20.175

4.7705

16.04

0.2033

2.4324

2.1589

1.712

5

0.8

3.4082

3.0407

3.3251

2.6954

2.7899

2.2373

2.1843

0.9

6.1551

4.9784

5.8855

3.9317

4.2714

2.7455

2.6302

0.95

11.704

7.8841

10.806

4.84

6.0539

2.3076

2.144

0.99

56.042

13.251

44.554

0.5647

6.7567

5.9966

4.7552

10

0.8

13.633

12.162

13.3

10.78

11.159

8.946

8.7336

0.9

24.62

19.913

23.542

15.726

17.085

10.979

10.518

0.95

46.818

31.537

43.226

19.36

24.216

9.23

8.5758

0.99

224.17

53.006

178.22

2.2591

27.027

23.986

19.02