Statistical Models and Methods for Reliability and Survival Analysis

Vincent Couallier is Associate Professor at Bordeaux Segalen University in France Léo Gerville-Réache is Associate Professor at Bordeaux 2 University in France. Catherine Huber-Carol is Professor Emeritus at Paris René Descartes University in France. Nikolaos Limnios is Professor at Compiègne University of Technology in France. Mounir Mesbah is Professor at University Pierre and Marie Curie in Paris, France.

Preface xv

Biography of Mikhail Stepanovitch Nikouline xvii
Vincent COUALLIER, Léo GERVILLE-RÉACHE, Catherine HUBER-CAROL, Nikolaos LIMNIOS and Mounir MESBAH

Part 1. Statistical Models and Methods 1

Chapter 1. Unidimensionality, Agreement and Concordance Probability 3
Zhezhen JIN and Mounir MESBAH

1.1. Introduction 3

1.2. From reliability to unidimensionality: CAC and curve 4

1.2.1. Classical unidimensional models for measurement 4

1.2.2. Reliability of an instrument: CAC 6

1.2.3. Unidimensionality of an instrument: BRC 9

1.3. Agreement between binary outcomes: the kappa coefficient 10

1.3.1. The kappa model 10

1.3.2. The kappa coefficient 10

1.3.3. Estimation of the kappa coefficient 10

1.4. Concordance probability 11

1.4.1. Relationship with Kendall’s τ measure 12

1.4.2. Relationship with Somer’s D measure 12

1.4.3. Relationship with ROC curve 13

1.5. Estimation and inference 14

1.6. Measure of agreement 14

1.7. Extension to survival data 15

1.7.1. Harrell’s c-index 15

1.7.2. Measure of discriminatory power 16

1.8. Discussion 17

1.9. Bibliography 18

Chapter 2. A Universal Goodness-of-Fit Test Based on Regression Techniques 21
Florence GEORGE and Sneh GULATI

2.1. Introduction 21

2.2. The Brain and Shapiro procedure for the exponential distribution 22

2.3. Applications of the Brain and Shapiro test 24

2.4. Small sample null distribution of the test statistic for specific distributions 25

2.5. Power studies 28

2.6. Some real examples 28

2.7. Conclusions 31

2.8. Acknowledgment 32

2.9. Bibliography 32

Chapter 3. Entropy-type Goodness-of-Fit Tests for Heavy-Tailed Distributions 33
Andreas MAKRIDES, Alex KARAGRIGORIOU and Filia VONTA

3.1. Introduction 33

3.2. The entropy test for heavy-tailed distributions 35

3.2.1. Development and asymptotic theory 35

3.2.2. Discussion 39

3.3. Simulation study 40

3.4. Conclusions 42

3.5. Bibliography 42

Chapter 4. Penalized Likelihood Methodology and Frailty Models 45
Emmanouil ANDROULAKIS, Christos KOUKOUVINOS and Filia VONTA

4.1. Introduction 45

4.2. Penalized likelihood in frailty models for clustered data 48

4.2.1. Gamma distributed frailty 52

4.2.2. Inverse Gaussian distributed frailty 52

4.2.3. Uniform distributed frailty 54

4.3. Simulation results 55

4.4. Concluding remarks 57

4.5. Bibliography 57

Chapter 5. Interactive Investigation of Statistical Regularities in Testing Composite Hypotheses of Goodness of Fit 61
Boris LEMESHKO, Stanislav LEMESHKO and Andrey ROGOZHNIKOV

5.1. Introduction 61

5.2. Distributions of the test statistics in the case of testing composite hypotheses 63

5.3. Testing composite hypotheses in “real-time” 68

5.4. Conclusions 73

5.5. Acknowledgment 73

5.6. Bibliography 73

Chapter 6. Modeling of Categorical Data 77
Henning LÄUTER

6.1. Introduction 77

6.2. Continuous conditional distributions 78

6.2.1. Conditional normal distribution 78

6.2.1.1. Estimation of parameters 78

6.2.2. More general continuous conditional distributions 81

6.2.2.1. Conditional distribution 82

6.2.2.2. Normal copula 83

6.3. Discrete conditional distributions 84

6.3.1. Parametric conditional distributions 84

6.3.2. Estimation of parameters 86

6.4. Goodness of fit 86

6.4.1. Distribution of ˆX2 87

6.5. Modeling of categorical data 88

6.5.1. Contingency tables 89

6.5.1.1. General tables 89

6.5.1.2. Further examples 93

6.6. Bibliography 93

Chapter 7. Within the Sample Comparison of Prediction Performance of Models and Submodels: Application to Alzheimer’s Disease 95
Catherine HUBER-CAROL, Shulamith T. GROSS and Annick ALPÉROVITCH

7.1. Introduction 95

7.2. Framework 96

7.2.1. General description of the data set and the models to be compared 96

7.2.2. Definition of the performance prediction criteria: IDI and BRI 96

7.3. Estimation of IDI and BRI 97

7.3.1. General estimating equations for IDI and BRI 98

7.3.2. Estimation of IDI and BRI in the logistic case 98

7.3.2.1. Asymptotics of IDI2/1 for logistic predictors 99

7.3.2.2. Asymptotics of BRI2/1 for logistic predictors 100

7.4. Simulation studies 102

7.4.1. First simulation 102

7.4.2. Second simulation: Gu and Pepe’s example 104

7.5. The three city study of Alzheimer’s disease 106

7.6. Conclusion 108

7.7. Bibliography 109

Chapter 8. Durbin–Knott Components and Transformations of the Cramér-von Mises Test 111
Gennady MARTYNOV

8.1. Introduction 111

8.2. Weighted Cramér-von Mises statistic 111

8.3. Examples of the Cramér-von Mises statistics 113

8.3.1. Classical Cramér-von Mises statistic 113

8.3.2. Anderson–Darling statistic 113

8.3.3. Cramér-von Mises statistic with the power weight function 114

8.4. Weighted parametric Cramér-von Mises statistic 114

8.4.1. Covariance functions of weighted parametric empirical process 114

8.4.2. Eigenvalues and eigenfunctions for weighted parametric Cramérvon Mises statistic 116

8.5. Transformations of the Cramér-von Mises statistic 117

8.5.1. Preliminary notes 117

8.5.2. Replacement of eigenvalues 118

8.5.3. Transformed statistics 119

8.6. Bibliography 122

Chapter 9. Conditional Inference in Parametric Models 125
Michel BRONIATOWSKI and Virgile CARON

9.1. Introduction and context 125

9.2. The approximate conditional density of the sample 127

9.2.1. Approximation of conditional densities 127

9.2.2. The proxy of the conditional density of the sample 129

9.2.3. Comments on implementation 131

9.3. Sufficient statistics and approximated conditional density 131

9.3.1. Keeping sufficiency under the proxy density 131

9.3.2. Rao–Blackwellization 132

9.4. Exponential models with nuisance parameters 135

9.4.1. Conditional inference in exponential families 135

9.4.2. Application of conditional sampling to MC tests 137

9.4.2.1. Context 137

9.4.2.2. Bimodal likelihood: testing the mean of a normal distribution in dimension 2 139

9.4.3. Estimation through conditional likelihood 140

9.5. Bibliography 142

Chapter 10. On Testing Stochastic Dominance by Exceedance, Precedence and Other Distribution-Free Tests, with Applications 145
Paul DEHEUVELS

10.1. Introduction 145

10.2. Results 148

10.2.1. The experimental data set 148

10.2.2. An application of the Wilcoxon–Mann–Whitney statistics 149

10.2.3. One-sided Kolmogorov-Smirnov tests 150

10.2.4. Precedence and Exceedance Tests. 152

10.3. Negative binomial limit laws 155

10.4. Conclusion 159

10.5. Bibliography 159

Chapter 11. Asymptotically Parameter-Free Tests for Ergodic Diffusion Processes 161
Yury A. KUTOYANTS and Li ZHOU

11.1. Introduction 161

11.2. Ergodic diffusion process and some limits 165

11.3. Shift parameter 168

11.4. Shift and scale parameters 172

11.5. Bibliography 175

Chapter 12. A Comparison of Homogeneity Tests for Different Alternative Hypotheses 177
Sergey POSTOVALOV and Petr PHILONENKO

12.1. Homogeneity tests 178

12.1.1. Tests for data without censoring 179

12.1.2. Tests for data with censoring 180

12.2. Alternative hypotheses 184

12.3. Power simulation 185

12.3.1. Power of tests without censoring 187

12.3.2. Power of tests with censoring 189

12.3.2.1. How does the distribution of censoring time affect the power of the test? 189

12.3.2.2. How does the censoring rate affect the power of the test? 191

12.4. Statistical inference 191

12.5. Acknowledgment 192

12.6. Bibliography 193

Chapter 13. Some Asymptotic Results for Exchangeably Weighted Bootstraps of the Empirical Estimator of a Semi-Markov Kernel with Applications 195
Salim BOUZEBDA and Nikolaos LIMNIOS

13.1. Introduction 195

13.2. Semi-Markov setting 197

13.3. Main results 201

13.4. Bootstrap for a multidimensional empirical estimator of a continuous-time semi-Markov kernel 205

13.5. Confidence intervals 208

13.6. Bibliography 210

Chapter 14. On Chi-Squared Goodness-of-Fit Test for Normality 213
Mikhail NIKULIN, Léo GERVILLE-RÉACHE and Xuan Quang TRAN

14.1. Chi–squared test for normality 213

14.2. Simulation study 221

14.3. Bibliography 226

Part 2. Statistical Models and Methods in Survival Analysis 229

Chapter 15. Estimation/Imputation Strategies for Missing Data in Survival Analysis 231
Elodie BRUNEL, Fabienne COMTE and Agathe GUILLOUX

15.1. Introduction 231

15.2. Model and strategies 233

15.2.1. Model assumptions 233

15.2.2. Strategy involving knowledge of ζ 234

15.2.3. Strategy involving knowledge of π 235

15.2.4. Estimation of ζ or π: logit or non-parametric regression 236

15.2.5. Computing the hazard estimators 236

15.2.6. Theoretical results 239

15.3. Imputation-based strategy 241

15.4. Numerical comparison 242

15.5. Proofs 244

15.6. Bibliography 251

Chapter 16. Non-Parametric Estimation of Linear Functionals of a Multivariate Distribution Under Multivariate Censoring with Applications 253
Olivier LOPEZ and Philippe SAINT-PIERRE

16.1. Introduction 253

16.2. Non-parametric estimation of the distribution 255

16.3. Asymptotic properties 257

16.4. Statistical applications of functionals 260

16.4.1. Dependence measures 260

16.4.2. Bootstrap 261

16.4.3. Linear regression 262

16.5. Illustration 263

16.6. Conclusion 264

16.7. Acknowledgment 264

16.8. Bibliography 264

Chapter 17. Kernel Estimation of Density from Indirect Observation 267
Valentin SOLEV

17.1. Introduction 267

17.1.1. Random partition 267

17.1.2. Indirect observation 268

17.1.3. Kernel density estimator 269

17.2. Density of random vector Λ(X) 271

17.3. Pseudo-kernel density estimator 273

17.3.1. Pointwise density estimation based on indirect data 273

17.3.2. Bias of the kernel estimator 274

17.3.3. Estimate of variance 276

17.4. Bibliography 279

Chapter 18. A Comparative Analysis of Some Chi-Square Goodness-of-Fit Tests for Censored Data 281
Ekaterina CHIMITOVA and Boris LEMESHKO

18.1. Introduction 281

18.2. Chi-square goodness-of-fit tests for censored data 283

18.2.1. NRR χ2 test 283

18.2.2. GPF χ2 test 284

18.3. The choice of grouping intervals 285

18.3.1. Equifrequent grouping (EFG) 289

18.3.2. Intervals with equal expected numbers of failures (EENFG) 289

18.3.3. Optimal grouping (OptG) 289

18.4. Empirical power study 290

18.5. Conclusions 293

18.6. Acknowledgment 294

18.7. Bibliography 294

Chapter 19. A Non-parametric Test for Comparing Treatments with Missing Data and Dependent Censoring 297
Amel MEZAOUER, Kamal BOUKHETALA and Jean-François DUPUY

19.1. Introduction 297

19.2. The proposed test statistic 299

19.3. Asymptotic distribution of the proposed test statistic 301

19.4. Acknowledgment 305

19.5. Appendix 306

19.6. Bibliography 309

Chapter 20. Group Sequential Tests for Treatment Effect with Covariates Adjustment through Simple Cross-Effect Models 311
Isaac Wu HONG-DAR

20.1. Introduction 311

20.2. Notations and models 313

20.3. Group sequential test 316

20.4. Discussion 318

20.5. Acknowledgment 318

20.6. Bibliography 318

Part 3. Reliability and Maintenance 321

Chapter 21. Optimal Maintenance in Degradation Processes 323
Waltraud KAHLE

21.1. Introduction 323

21.2. The degradation model 324

21.3. Optimal replacement after an inspection 326

21.4. The simulation of degradation processes 327

21.5. Shape of cost functions and optimal δ and a 329

21.6. Incomplete preventive maintenance 330

21.7. Bibliography 333

Chapter 22. Planning Accelerated Destructive Degradation Tests with Competing Risks 335
Ying SHI and William Q. MEEKER

22.1. Introduction 336

22.1.1. Background 336

22.1.2. Motivation: adhesive bond C 336

22.1.3. Related literature 337

22.1.4. Overview 338

22.2. Degradation models with competing risks 338

22.2.1. Accelerated degradation model for the primary response 338

22.2.2. Accelerated degradation model for the competing response 339

22.2.3. Degradation models for adhesive bond C 339

22.2.4. Degradation distribution and quantiles 340

22.3. Failure-time distribution with competing risks 341

22.3.1. Relationship between degradation and failure 341

22.3.2. Failure-time distribution and quantiles 342

22.4. Test planning with competing risks 342

22.4.1. ADDT planning information 342

22.4.2. Criterion for ADDT planning with competing risks 343

22.5. ADDT plans with competing risks 344

22.5.1. Initial optimum ADDT plan with competing risks 344

22.5.2. Constrained optimum ADDT plan with competing risks 348

22.5.3. General equivalence theorem 348

22.5.4. Compromise ADDT plan with competing risks 350

22.6. Monte Carlo simulation to evaluate test plans 352

22.7. Conclusions and extensions 353

22.8. Appendix: technical details 354

22.8.1. The Fisher information matrix for ADDT with competing risks 354

22.8.2. Large-sample approximate variance of ht (tp) and tp 355

22.9. Bibliography 355

Chapter 23. A New Goodness-of-Fit Test for Shape-Scale Families 357
Vilijandas BAGDONAVIČIUS

23.1. Introduction 357

23.2. The test statistic 358

23.3. The asymptotic distribution of the test statistic 359

23.4. The test 364

23.5. Weibull distribution 364

23.6. Loglogistic distribution 365

23.7. Lognormal distribution 366

23.8. Bibliography 367

Chapter 24. Time-to-Failure of Markov-Modulated Gamma Process with Application to Replacement Policies 369
Christian PAROISSIN and Landy RABEHASAINA

24.1. Introduction 369

24.2. Degradation model 370

24.2.1. Covariate process 370

24.2.2. Degradation process 371

24.3. Time-to-failure distribution 371

24.3.1. Case of a non-modulated gamma process 372

24.3.2. Case of a Markov-modulated gamma process 373

24.3.3. Stochastic comparison 374

24.4. Replacement policies 376

24.4.1. Block replacement policy 377

24.4.2. Age replacement policy 379

24.5. Conclusion 381

24.6. Acknowledgment 381

24.7. Bibliography 382

Chapter 25. Calculation of the Redundant Structure Reliability for Agingtype Elements 383
Alexandr ANTONOV, Alexandr PLYASKIN and Khizri TATAEV

25.1. Introduction 383

25.2. The operation process of the renewal and repaired products 384

25.3. The model of the geometric process 386

25.4. Task solution 387

25.5. Conclusion 389

25.6. Bibliography 390

Chapter 26. On Engineering Risks of Complex Hierarchical Systems Analysis 391
Vladimir RYKOV

26.1. Introduction 391

26.2. Risk definition and measurement 392

26.3. Engineering risk 393

26.4. Risk characteristics for general model calculation 395

26.4.1. Lifelength and appropriate loss size CDF 395

26.4.2. Probability of risk event evolution 396

26.4.3. Lifelength and loss moments 397

26.4.4. Mostly dangerous paths of risk event evolution and sensitivity analysis 399

26.5. Risk analysis for short-time risk models 400

26.6. Conclusion 402

26.7. Bibliography 402

List of Authors 405

Index 409