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Correction to: Comparison of joint control schemes for multivariate normal i.i.d. output

  • Manuel Cabral MoraisEmail author
  • Wolfgang Schmid
  • Patrícia Ferreira Ramos
  • Taras Lazariv
  • António Pacheco
  • Ivan Semeniuk
Correction
  • 113 Downloads

1 Correction to: AStA Advances in Statistical Analysis  https://doi.org/10.1007/s10182-018-00331-3

In the original paper (Morais et al. 2018), we incorrectly stated that
$$\begin{aligned} T^{(1)} = n \, \left( \bar{\mathbf {X}}-\varvec{\mu }_0 \right) ^\top \, \varvec{\varSigma }_0^{-1} \, \left( \bar{\mathbf {X}}-\varvec{\mu }_0 \right) \sim \chi ^2_p (\lambda ), \end{aligned}$$
where \(\chi ^2_p (\lambda )\) represents the noncentral Chi-square distribution with p degrees of freedom and noncentrality parameter \(\lambda = (\varvec{\varSigma }_0^{1/2} \, \varvec{\delta })^\top \, \varvec{\varSigma }^{-1} \,(\varvec{\varSigma }_0^{1/2} \, \varvec{\delta })\). This statement is only valid when \(\varvec{\varSigma }= \varvec{\varSigma }_0\), according to Anderson (1958, p. 55), Muirhead (1982, p. 26, Theorem 1.4.1) and Mathai and Provost (1992, p. 201, Corollary 5.1.3a). In this particular case, we have \(\lambda = \varvec{\delta }^\top \times \varvec{\delta }\).
Table 1

Control limits of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to \(ARL^{(i)}(\mathbf{0},\varvec{\varSigma }_0)=ARL^{\star }=500\)

p

\(n=p+5\)

Scheme i

\(\beta ^{(i)}\)

\(UCL_{\varvec{\mu }}^{(i)}\)

\(UCL_{\varvec{\varSigma }}^{(i)}\)

2

7

1

0.001000501

13.814510

218.847114

3

8

1

0.001000501

16.265177

2221.257632

4

9

1

0.001000501

18.465717

23246.080493

Table 2

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(2,7)\), scenarios 1 and 2

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0.3)

(1.05, 0.3)

(1.2, 0.3)

(1.5, 0.3)

(2, 0.3)

1

0

500.000

472.615

371.851

142.207

30.362

7.111

 

458.104

325.220

108.945

24.010

5.989

 

499.513

457.384

325.012

108.899

24.045

5.984

 

0.5

233.038

228.404

206.498

113.000

29.164

7.067

 

218.671

170.238

72.951

20.046

5.579

 

233.299

218.760

170.425

73.031

20.078

5.577

 

2

4.295

4.371

4.681

5.858

7.304

5.162

 

4.265

4.147

3.762

3.128

2.298

 

4.294

4.267

4.155

3.753

3.124

2.298

 
 

(1, 0.3)

(1.01, 0.18)

(1.05, 0.18)

(1.2, 0.18)

(1.5, 0.18)

(2, 0.18)

2

0

500.000

391.981

302.119

113.203

25.239

6.305

 

322.194

233.364

82.977

19.759

5.300

 

499.673

322.206

233.731

82.877

19.734

5.307

 

0.5

233.038

195.748

174.162

91.928

24.313

6.267

 

190.461

146.818

61.745

17.160

4.994

 

232.719

190.699

146.667

61.839

17.121

4.992

 

2

4.295

3.675

3.926

4.884

6.096

4.521

 

4.276

4.148

3.728

3.046

2.204

 

4.295

4.276

4.151

3.729

3.048

2.201

 
 

(1, 0.3)

(1.01, 0.303)

(1.05, 0.303)

(1.2, 0.303)

(1.5, 0.303)

(2, 0.303)

2

0

500.000

475.332

374.274

143.267

30.548

7.139

 

462.174

328.026

109.777

24.151

6.012

 

500.109

460.423

327.600

109.710

24.171

6.018

 

0.5

233.038

195.748

174.162

91.928

24.313

6.267

 

219.241

170.769

73.267

20.139

5.598

 

233.022

218.933

170.810

73.240

20.163

5.593

 

2.0

4.295

3.675

3.926

4.884

6.096

4.521

 

4.264

4.147

3.763

3.130

2.301

 

4.292

4.257

4.146

3.767

3.134

2.300

 
 

(1, 0.3)

(1.01, 0.315)

(1.05, 0.315)

(1.2, 0.315)

(1.5, 0.315)

(2, 0.315)

 

0

500.000

486.581

384.357

147.718

31.331

7.258

 

478.702

339.499

113.211

24.735

6.107

 

499.871

479.215

339.559

113.126

24.755

6.120

 

0.5

233.038

233.362

211.654

116.791

30.071

7.214

 

221.425

172.841

74.543

20.519

5.678

 

232.515

221.501

173.023

74.568

20.496

5.674

 

2.0

4.295

4.464

4.781

5.990

7.478

5.266

 

4.264

4.147

3.767

3.140

2.313

 

4.298

4.264

4.151

3.760

3.139

2.315

 
Table 3

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(2,7)\), scenarios 3 and 4

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0)

(1.05, 0)

(1.2, 0)

(1.5, 0)

(2, 0)

3

0

500.000

351.981

268.944

100.273

22.926

5.925

 

201.873

153.232

61.808

16.648

4.836

 

499.623

201.433

152.920

61.686

16.666

4.833

 

0.5

233.038

166.314

148.612

80.071

21.999

5.886

 

144.557

113.341

50.264

14.947

4.604

 

232.702

144.393

113.151

50.187

14.934

4.609

 

2

4.295

2.776

2.954

3.646

4.687

3.939

 

4.314

4.170

3.704

2.974

2.130

 

4.298

4.318

4.165

3.702

2.976

2.127

 
 

(1, 0)

\((1.01, -0.09)\)

\((1.05,-0.09)\)

\((1.2,-0.09)\)

\((1.5,-0.09)\)

\((2,-0.09)\)

4

0

500.000

483.532

380.878

145.232

29.386

7.140

 

453.318

323.716

109.812

24.345

6.059

 

500.118

451.532

321.468

108.046

24.532

6.011

 

0.5

233.038

216.670

198.542

112.530

57.114

7.094

 

240.011

184.891

77.041

20.641

5.664

 

236.523

239.891

183.595

77.147

20.431

5.633

 

2

4.295

3.703

3.985

5.092

6.678

5.048

 

4.363

4.234

3.819

3.151

2.301

 

4.292

4.407

4.178

3.845

3.183

2.258

 
 

(1, 0)

(1.01, 0.01)

(1.05, 0.01)

(1.2, 0.01)

(1.5, 0.01)

(2, 0.01)

4

0

500.000

472.748

371.961

142.244

30.366

7.111

 

458.046

325.203

108.955

24.014

5.990

 

499.368

454.630

321.563

110.588

23.346

6.010

 

0.5

233.038

216.670

198.542

112.530

29.386

7.094

 

216.174

168.568

72.552

20.011

5.577

 

232.441

217.611

168.916

72.547

20.437

5.508

 

2

4.295

4.450

4.762

5.947

7.376

5.175

 

4.255

4.138

3.757

3.126

2.299

 

4.241

4.303

4.100

3.812

3.142

2.322

 
 

(1, 0)

(1.01, 0.1)

(1.05, 0.1)

(1.2, 0.1)

(1.5, 0.1)

(2, 0.1)

 

0

500.000

486.117

383.023

145.953

30.781

7.146

 

452.183

323.340

110.011

24.424

6.076

 

500.643

459.877

322.460

108.414

24.533

6.111

 

0.5

233.038

245.831

220.538

117.637

29.636

7.105

 

193.599

153.635

69.420

19.954

5.631

 

232.188

192.446

154.350

70.510

19.651

5.578

 

2

4.295

5.195

5.533

6.783

8.043

5.309

 

4.175

4.068

3.715

3.121

2.315

 

4.276

4.223

4.100

3.732

3.143

2.301

 
Table 4

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(3,8)\), scenarios 1 and 2

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0.3)

(1.05, 0.3)

(1.2, 0.3)

(1.5, 0.3)

(2, 0.3)

1

0

500.000

467.837

351.115

110.813

19.156

4.239

 

451.161

304.071

86.340

15.820

3.727

 

499.682

451.047

303.713

86.361

15.838

3.727

 

0.5

207.363

203.173

182.179

89.087

18.592

4.223

 

193.297

146.773

56.875

13.401

3.517

 

207.522

193.365

147.025

56.945

13.408

3.512

 

2

2.609

2.652

2.828

3.520

4.457

3.184

 

2.598

2.553

2.400

2.101

1.631

 

2.610

2.597

2.553

2.403

2.102

1.633

 
 

(1, 0.3)

(1.01, 0.18)

(1.05, 0.18)

(1.2, 0.18)

(1.5, 0.18)

(2, 0.18)

2

0

500.000

317.746

229.017

71.057

13.774

3.504

 

244.582

170.013

53.398

11.331

3.086

 

500.296

244.187

170.391

53.405

11.345

3.088

 

0.5

207.363

150.968

130.612

59.566

13.422

3.492

 

147.492

109.853

41.210

10.122

2.954

 

207.657

147.616

109.988

41.130

10.108

2.955

 

2

2.609

2.051

2.171

2.642

3.304

2.586

 

2.526

2.477

2.307

1.981

1.526

 

2.608

2.526

2.477

2.308

1.981

1.525

 
 

(1, 0.3)

(1.01, 0.303)

(1.05, 0.303)

(1.2, 0.303)

(1.5, 0.303)

(2, 0.303)

2

0

500.000

473.017

355.603

112.402

19.365

4.266

 

458.692

309.012

87.573

15.985

3.750

 

499.597

458.905

308.547

87.599

15.959

3.749

 

0.5

207.363

204.727

183.821

90.195

18.791

4.250

 

194.296

147.680

57.361

13.516

3.537

 

207.281

194.215

147.932

57.356

13.513

3.539

 

2

2.609

2.669

2.846

3.545

4.492

3.204

 

2.599

2.555

2.402

2.104

1.635

 

2.613

2.599

2.554

2.400

2.104

1.635

 
 

(1, 0.3)

(1.01, 0.315)

(1.05, 0.315)

(1.2, 0.315)

(1.5, 0.315)

(2, 0.315)

 

0

500.000

494.460

374.403

119.179

20.253

4.379

 

490.150

329.722

92.771

16.680

3.845

 

499.809

489.982

330.250

92.756

16.669

3.849

 

0.5

207.363

211.042

190.558

94.873

19.636

4.362

 

198.121

151.228

59.342

13.992

3.619

 

207.283

198.142

151.276

59.223

13.995

3.618

 

2

2.609

2.736

2.920

3.644

4.632

3.286

 

2.607

2.563

2.412

2.117

1.648

 

2.604

2.605

2.566

2.412

2.119

1.648

 
Table 5

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(3,8)\), scenarios 3 and 4

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0)

(1.05, 0)

(1.2, 0)

(1.5, 0)

(2, 0)

3

0

500.000

247.574

176.316

55.399

11.520

3.167

 

128.517

94.872

35.122

8.829

2.718

 

500.038

128.506

94.909

35.137

8.838

2.718

 

0.5

207.363

105.674

92.757

45.046

11.157

3.152

 

94.980

72.604

29.619

8.145

2.628

 

207.211

94.790

72.623

29.618

8.141

2.630

 

2

2.609

1.397

1.453

1.685

2.108

2.033

 

2.428

2.374

2.188

1.857

1.439

 

2.612

2.425

2.375

2.188

1.856

1.439

 
 

(1, 0)

\((1.01,-0.09)\)

\((1.05,-0.09)\)

\((1.2,-0.09)\)

\((1.5,-0.09)\)

\((2,-0.09)\)

4

0

500.000

495.342

373.281

116.792

19.647

4.274

 

447.437

305.625

89.144

16.440

3.827

 

499.253

448.187

305.619

88.184

16.089

3.747

 

0.5

207.363

176.419

164.114

88.556

18.953

4.256

 

232.951

173.260

63.558

14.250

3.627

 

206.973

232.406

172.166

62.235

13.864

3.547

 

2

2.609

1.955

2.093

2.667

3.684

3.042

 

2.604

2.555

2.388

2.080

1.618

 

2.605

2.605

2.554

2.389

2.086

1.623

 
 

(1, 0)

(1.01, 0.01)

(1.05, 0.01)

(1.2, 0.01)

(1.5, 0.01)

(2, 0.01)

4

0

500.000

468.149

351.365

110.880

19.162

4.239

 

451.083

304.072

86.370

15.827

3.728

 

500.360

451.202

303.275

86.450

15.828

3.726

 

0.5

207.363

206.271

184.454

89.503

18.606

4.224

 

188.882

143.928

56.303

13.365

3.516

 

207.286

188.500

143.929

56.326

13.375

3.515

 

2

2.609

2.740

2.920

3.625

4.547

3.201

 

2.597

2.554

2.401

2.104

1.634

 

2.610

2.598

2.554

2.398

2.100

1.633

 
 

(1, 0)

(1.01, 0.1)

(1.05, 0.1)

(1.2, 0.1)

(1.5, 0.1)

(2, 0.1)

 

0

500.000

497.829

375.427

117.480

19.715

4.280

 

439.085

301.424

88.852

16.472

3.835

 

500.434

438.885

302.352

88.018

16.129

3.759

 

0.5

207.363

238.022

209.891

96.908

19.201

4.265

 

151.719

119.996

52.157

13.448

3.593

 

207.397

152.066

121.192

52.847

13.319

3.529

 

2

2.609

3.638

3.855

4.663

5.396

3.352

 

2.600

2.560

2.421

2.140

1.670

 

2.609

2.601

2.559

2.416

2.126

1.647

 
Table 6

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(4,9)\), scenarios 1 and 2

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0.3)

(1.05, 0.3)

(1.2, 0.3)

(1.5, 0.3)

(2, 0.3)

1

0

500.000

463.386

332.387

87.848

12.892

2.860

 

444.680

285.582

69.874

11.023

2.600

 

500.691

446.563

286.118

69.930

11.013

2.596

 

0.5

187.575

183.761

163.557

71.725

12.604

2.853

 

173.864

129.284

45.864

9.500

2.482

 

187.590

173.998

129.456

45.861

9.517

2.478

 

2

1.896

1.923

2.035

2.486

3.111

2.239

 

1.890

1.871

1.798

1.629

1.327

 

1.895

1.890

1.873

1.797

1.628

1.327

 
 

(1, 0.3)

(1.01, 0.18)

(1.05, 0.18)

(1.2, 0.18)

(1.5, 0.18)

(2, 0.18)

2

0

500.000

256.507

173.950

46.421

8.334

2.295

 

191.322

129.157

36.264

7.149

2.094

 

499.523

193.056

129.078

36.241

7.139

2.094

 

0.5

187.575

119.726

100.560

40.096

8.181

2.290

 

116.536

84.712

28.767

6.518

2.029

 

187.749

117.181

84.742

28.793

6.516

2.028

 

2

1.896

1.466

1.531

1.793

2.172

1.773

 

1.793

1.772

1.690

1.508

1.236

 

1.895

1.791

1.772

1.690

1.508

1.236

 
 

(1, 0.3)

(1.01, 0.303)

(1.05, 0.303)

(1.2, 0.303)

(1.5, 0.303)

(2, 0.303)

2

0

500.000

470.825

338.644

89.656

13.079

2.881

 

457.744

292.599

71.228

11.178

2.615

 

500.037

456.821

292.922

71.382

11.184

2.617

 

0.5

187.575

185.670

165.616

73.011

12.784

2.874

 

175.557

130.543

46.418

9.612

2.495

 

187.600

175.226

130.580

46.386

9.616

2.499

 

2

1.896

1.936

2.049

2.506

3.141

2.255

 

1.893

1.873

1.801

1.632

1.329

 

1.896

1.890

1.872

1.799

1.633

1.329

 
 

(1, 0.3)

(1.01, 0.315)

(1.05, 0.315)

(1.2, 0.315)

(1.5, 0.315)

(2, 0.315)

 

0

500.000

501.649

365.071

97.490

13.882

2.972

 

498.745

321.287

77.290

11.837

2.693

 

500.276

500.087

321.307

77.289

11.825

2.696

 

0.5

187.575

193.402

174.062

78.500

13.556

2.964

 

180.064

135.230

48.800

10.085

2.564

 

187.509

180.579

135.258

48.749

10.089

2.563

 

2

1.896

1.990

2.109

2.589

3.262

2.321

 

1.903

1.883

1.812

1.646

1.341

 

1.896

1.903

1.882

1.812

1.646

1.341

 
Table 7

ARL values of the joint scheme of type 1 for \(\varvec{\mu }\) and \(\varvec{\varSigma }\), with in-control ARL equal to 500 (listed, for each value of \(\delta \), in order corresponding to the value in the original paper (Morais et al. 2018), the corrected value, and the Monte Carlo estimate)—\((p,n)=(4,9)\), scenarios 3 and 4

\(\delta \)

\((\sigma _0^2,\rho _0)\)

  

\((\sigma ^2,\rho )\)

  

Scenario

 

(1, 0.3)

(1.01, 0)

(1.05, 0)

(1.2, 0)

(1.5, 0)

(2, 0)

3

0

500.000

170.359

114.706

32.154

6.550

2.042

 

89.746

64.382

21.912

5.320

1.833

 

499.763

89.918

64.331

21.932

5.323

1.834

 

0.5

187.575

68.099

58.978

26.613

6.383

2.035

 

66.781

49.844

18.842

4.986

1.790

 

187.264

66.889

49.833

18.841

4.984

1.792

 

2

1.896

1.071

1.088

1.163

1.329

1.354

 

1.646

1.624

1.544

1.379

1.164

 

1.895

1.647

1.624

1.544

1.379

1.165

 
 

(1, 0)

\((1.01,-0.09)\)

\((1.05,-0.09)\)

\((1.2,-0.09)\)

\((1.5,-0.09)\)

\((2,-0.09)\)

4

0

500.000

514.134

372.369

96.600

13.443

2.894

 

446.488

296.250

75.280

11.860

2.707

 

499.749

449.300

294.197

73.383

11.401

2.625

 

0.5

187.575

139.927

133.658

71.334

28.775

13.037

 

228.766

166.073

54.425

10.475

2.595

 

187.682

228.900

164.145

52.586

9.989

2.507

 

2

1.896

1.324

1.393

1.704

2.351

2.099

 

1.831

1.810

1.735

1.576

1.303

 

1.897

1.832

1.812

1.746

1.592

1.312

 
 

(1, 0)

(1.01, 0.01)

(1.05, 0.01)

(1.2, 0.01)

(1.5, 0.01)

(2, 0.01)

4

0

500.000

463.914

332.797

87.939

12.898

2.860

 

447.056

286.646

69.928

11.039

2.600

 

499.801

445.014

286.189

69.884

11.030

2.595

 

0.5

187.575

188.466

166.946

72.224

12.617

2.853

 

168.345

125.820

45.239

9.477

2.481

 

187.439

167.922

126.011

45.367

9.480

2.479

 

2

1.896

2.007

2.124

2.589

3.202

2.254

 

1.897

1.877

1.805

1.635

1.330

 

1.894

1.898

1.876

1.802

1.632

1.329

 
 

(1, 0)

(1.01, 0.1)

(1.05, 0.1)

(1.2, 0.1)

(1.5, 0.1)

(2, 0.1)

 

0

500.000

512.016

371.058

96.591

13.468

2.897

 

425.455

283.917

73.883

11.785

2.701

 

500.940

427.935

284.224

72.983

11.388

2.625

 

0.5

187.575

235.446

204.197

81.598

13.217

2.891

 

122.049

96.785

41.089

9.695

2.560

 

187.337

122.856

98.249

41.725

9.527

2.496

 

2

1.896

2.934

3.092

3.677

4.077

2.391

 

1.948

1.930

1.863

1.697

1.372

 

1.896

1.948

1.927

1.854

1.674

1.346

 
When \(\varvec{\varSigma }\ne \varvec{\varSigma }_0\), we can capitalize on various series representations for the c.d.f. of \(T^{(1)}\), such as the one mentioned by Mathai and Provost (1992, p. 95, Theorem 4.12b.1), truncate such series, and provide approximations to the c.d.f. of \(T^{(1)}\). Instead, we favoured the representation of the quadratic form \(T^{(1)}\) as in Mathai and Provost (1992, p. 29, formula (3.1a.4)) or Duchesne and Micheaux (2010). By doing so, this control statistic can be written as a weighted sum of independent noncentral Chi-square random variables (Mathai and Provost 1992, p. 29),
$$\begin{aligned} T^{(1)} \, \sim \, \sum \limits _{i=1}^{p}\lambda _i \, \chi ^2_1 (\tau _i), \end{aligned}$$
where \(\lambda _i, \, i=1,\ldots ,p\), are the eigenvalues of the matrix
$$\begin{aligned} \varvec{\varSigma }^{1/2} \varvec{\varSigma }_0^{-1} \varvec{\varSigma }^{1/2}; \end{aligned}$$
\(\tau _i\) is the square of the ith element of vector
$$\begin{aligned} \varvec{U}^\top \varvec{\varSigma }^{-1/2} \varvec{\varSigma }_0^{1/2} \varvec{\delta }; \end{aligned}$$
\(\varvec{U}\) is the orthogonal matrix such that
$$\begin{aligned} \varvec{U}^\top \varvec{\varSigma }^{1/2} \varvec{\varSigma }_0^{-1} \varvec{\varSigma }^{1/2} \varvec{U} = \Lambda = \hbox {diag}(\lambda _1, \ldots , \lambda _p). \end{aligned}$$
Unfortunately, the incorrection we mentioned was only detected after the publication of Morais et al. (2018). The authors apologize for it and remind the reader that the correct marginal distribution of \(T^{(1)}\), when \(\varvec{\varSigma }\ne \varvec{\varSigma }_0\), ought to be considered in the statement of Theorem 1 in the original publication.

Fortunately, this had no bearing on the control limits of the joint scheme of type 1. However, the out-of-control ARL values referring to this joint scheme and to \(\varvec{\varSigma }\ne \varvec{\varSigma }_0\), in tables 2–7 of Morais et al. (2018), had to be recalculated. They were obtained by taking advantage of the package CompQuadForm (Micheaux 2017) for the R statistical software (R Core Team 2013); the distribution function of the quadratic form in normal variables \(T^{(1)}\) was computed using Imhof’s method (1961), with absolute and relative accuracies equal to \(10^{-6}\).

The reader must keep in mind that the out-of-control ARL values in Tables 2, 3, 4, 5, 6 and 7 refer to the joint schemes of type 1 with the parameters in Table 1 taken from Morais et al. (2018, Table 1).

Moreover, the out-of-control ARL values are listed in Tables 2, 3, 4, 5, 6 and 7 in order corresponding to the value in the original paper, the corrected value (in italic), and the corresponding estimate (obtained by Monte Carlo simulation with \(10^6\) replications), for each value of \(\delta \).

The correct out-of-control ARL values when \(\varvec{\varSigma }\ne \varvec{\varSigma }_0\) in Tables 2, 3, 4, 5, 6 and 7 are rarely larger than the ones in the original paper. In this particular case, the values are set in bold, but they are still smaller than the ones of the two competing joint schemes described in Morais et al. (2018). Furthermore, the ARL reductions are in some cases substantial.

Consequently, the correct ARL values we obtained serve to increase rather than to lessen our belief that the joint scheme 1 has the best overall ARL performance, as we mentioned in the abstract and the conclusions and recommendations of Morais et al. (2018). Unsurprisingly, the ARL profiles of figures 1–4 of the original paper were not redrawn.

Notes

References

  1. Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Wiley, New York (1958)zbMATHGoogle Scholar
  2. Duchesne, P., Micheaux, P.: Computing the distribution of quadratic forms: further comparisons between the Liu–Tang–Zhang approximation and exact methods. Comput. Stat. Data Anal. 54, 858–862 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Imhof, J.P.: Computing the distribution of quadratic forms in normal variables. Biometrika 48, 419–426 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Mathai, A.M., Provost, S.B.: Quadratic Forms in Random Variables. Dekker, New York (1992)zbMATHGoogle Scholar
  5. Micheaux, P.: Package ‘ompQuadForm’ (2017). https://cran.r-project.org/web/packages/CompQuadForm/index.html. R package version 1.4.3
  6. Morais, M.C., Schmid, W., Ramos, P.F., Lazariv, T., Pacheco, A.: Comparison of joint control schemes for multivariate normal i.i.d. output. AStA Adv. Stat. Anal. 12, 1–31 (2018).  https://doi.org/10.1007/s10182-018-00331-3 Google Scholar
  7. Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, Hoboken (1982)CrossRefzbMATHGoogle Scholar
  8. R Core Team: R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2013). http://www.R-project.org/

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior TécnicoULisboaLisbonPortugal
  2. 2.Department of StatisticsEuropean University ViadrinaFrankfurt (Oder)Germany
  3. 3.CEMATUniversidade Autónoma de LisboaLisbonPortugal
  4. 4.Center of Information Services and High Performance ComputingTU DresdenDresdenGermany

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