Statistical Models and Methods for Reliability and Survival Analysis

Vincent Couallier is Associate Professor at Bordeaux Segalen University in FranceLéo 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 xvBiography of Mikhail Stepanovitch Nikouline xviiVincent COUALLIER, Léo GERVILLE-RÉACHE, Catherine HUBER-CAROL, Nikolaos LIMNIOS and Mounir MESBAHPart 1. Statistical Models and Methods 1Chapter 1. Unidimensionality, Agreement and Concordance Probability 3Zhezhen JIN and Mounir MESBAH1.1. Introduction 31.2. From reliability to unidimensionality: CAC and curve 41.2.1. Classical unidimensional models for measurement 41.2.2. Reliability of an instrument: CAC 61.2.3. Unidimensionality of an instrument: BRC 91.3. Agreement between binary outcomes: the kappa coefficient 101.3.1. The kappa model 101.3.2. The kappa coefficient 101.3.3. Estimation of the kappa coefficient 101.4. Concordance probability 111.4.1. Relationship with Kendall’s τ measure 121.4.2. Relationship with Somer’s D measure 121.4.3. Relationship with ROC curve 131.5. Estimation and inference 141.6. Measure of agreement 141.7. Extension to survival data 151.7.1. Harrell’s c-index 151.7.2. Measure of discriminatory power 161.8. Discussion 171.9. Bibliography 18Chapter 2. A Universal Goodness-of-Fit Test Based on Regression Techniques 21Florence GEORGE and Sneh GULATI2.1. Introduction 212.2. The Brain and Shapiro procedure for the exponential distribution 222.3. Applications of the Brain and Shapiro test 242.4. Small sample null distribution of the test statistic for specific distributions 252.5. Power studies 282.6. Some real examples 282.7. Conclusions 312.8. Acknowledgment 322.9. Bibliography 32Chapter 3. Entropy-type Goodness-of-Fit Tests for Heavy-Tailed Distributions 33Andreas MAKRIDES, Alex KARAGRIGORIOU and Filia VONTA3.1. Introduction 333.2. The entropy test for heavy-tailed distributions 353.2.1. Development and asymptotic theory 353.2.2. Discussion 393.3. Simulation study 403.4. Conclusions 423.5. Bibliography 42Chapter 4. Penalized Likelihood Methodology and Frailty Models 45Emmanouil ANDROULAKIS, Christos KOUKOUVINOS and Filia VONTA4.1. Introduction 454.2. Penalized likelihood in frailty models for clustered data 484.2.1. Gamma distributed frailty 524.2.2. Inverse Gaussian distributed frailty 524.2.3. Uniform distributed frailty 544.3. Simulation results 554.4. Concluding remarks 574.5. Bibliography 57Chapter 5. Interactive Investigation of Statistical Regularities in Testing Composite Hypotheses of Goodness of Fit 61Boris LEMESHKO, Stanislav LEMESHKO and Andrey ROGOZHNIKOV5.1. Introduction 615.2. Distributions of the test statistics in the case of testing composite hypotheses 635.3. Testing composite hypotheses in “real-time” 685.4. Conclusions 735.5. Acknowledgment 735.6. Bibliography 73Chapter 6. Modeling of Categorical Data 77Henning LÄUTER6.1. Introduction 776.2. Continuous conditional distributions 786.2.1. Conditional normal distribution 786.2.1.1. Estimation of parameters 786.2.2. More general continuous conditional distributions 816.2.2.1. Conditional distribution 826.2.2.2. Normal copula 836.3. Discrete conditional distributions 846.3.1. Parametric conditional distributions 846.3.2. Estimation of parameters 866.4. Goodness of fit 866.4.1. Distribution of ˆX2 876.5. Modeling of categorical data 886.5.1. Contingency tables 896.5.1.1. General tables 896.5.1.2. Further examples 936.6. Bibliography 93Chapter 7. Within the Sample Comparison of Prediction Performance of Models and Submodels: Application to Alzheimer’s Disease 95Catherine HUBER-CAROL, Shulamith T. GROSS and Annick ALPÉROVITCH7.1. Introduction 957.2. Framework 967.2.1. General description of the data set and the models to be compared 967.2.2. Definition of the performance prediction criteria: IDI and BRI 967.3. Estimation of IDI and BRI 977.3.1. General estimating equations for IDI and BRI 987.3.2. Estimation of IDI and BRI in the logistic case 987.3.2.1. Asymptotics of IDI2/1 for logistic predictors 997.3.2.2. Asymptotics of BRI2/1 for logistic predictors 1007.4. Simulation studies 1027.4.1. First simulation 1027.4.2. Second simulation: Gu and Pepe’s example 1047.5. The three city study of Alzheimer’s disease 1067.6. Conclusion 1087.7. Bibliography 109Chapter 8. Durbin–Knott Components and Transformations of the Cramér-von Mises Test 111Gennady MARTYNOV8.1. Introduction 1118.2. Weighted Cramér-von Mises statistic 1118.3. Examples of the Cramér-von Mises statistics 1138.3.1. Classical Cramér-von Mises statistic 1138.3.2. Anderson–Darling statistic 1138.3.3. Cramér-von Mises statistic with the power weight function 1148.4. Weighted parametric Cramér-von Mises statistic 1148.4.1. Covariance functions of weighted parametric empirical process 1148.4.2. Eigenvalues and eigenfunctions for weighted parametric Cramérvon Mises statistic 1168.5. Transformations of the Cramér-von Mises statistic 1178.5.1. Preliminary notes 1178.5.2. Replacement of eigenvalues 1188.5.3. Transformed statistics 1198.6. Bibliography 122Chapter 9. Conditional Inference in Parametric Models 125Michel BRONIATOWSKI and Virgile CARON9.1. Introduction and context 1259.2. The approximate conditional density of the sample 1279.2.1. Approximation of conditional densities 1279.2.2. The proxy of the conditional density of the sample 1299.2.3. Comments on implementation 1319.3. Sufficient statistics and approximated conditional density 1319.3.1. Keeping sufficiency under the proxy density 1319.3.2. Rao–Blackwellization 1329.4. Exponential models with nuisance parameters 1359.4.1. Conditional inference in exponential families 1359.4.2. Application of conditional sampling to MC tests 1379.4.2.1. Context 1379.4.2.2. Bimodal likelihood: testing the mean of a normal distribution in dimension 2 1399.4.3. Estimation through conditional likelihood 1409.5. Bibliography 142Chapter 10. On Testing Stochastic Dominance by Exceedance, Precedence and Other Distribution-Free Tests, with Applications 145Paul DEHEUVELS10.1. Introduction 14510.2. Results 14810.2.1. The experimental data set 14810.2.2. An application of the Wilcoxon–Mann–Whitney statistics 14910.2.3. One-sided Kolmogorov-Smirnov tests 15010.2.4. Precedence and Exceedance Tests. 15210.3. Negative binomial limit laws 15510.4. Conclusion 15910.5. Bibliography 159Chapter 11. Asymptotically Parameter-Free Tests for Ergodic Diffusion Processes 161Yury A. KUTOYANTS and Li ZHOU11.1. Introduction 16111.2. Ergodic diffusion process and some limits 16511.3. Shift parameter 16811.4. Shift and scale parameters 17211.5. Bibliography 175Chapter 12. A Comparison of Homogeneity Tests for Different Alternative Hypotheses 177Sergey POSTOVALOV and Petr PHILONENKO12.1. Homogeneity tests 17812.1.1. Tests for data without censoring 17912.1.2. Tests for data with censoring 18012.2. Alternative hypotheses 18412.3. Power simulation 18512.3.1. Power of tests without censoring 18712.3.2. Power of tests with censoring 18912.3.2.1. How does the distribution of censoring time affect the power of the test? 18912.3.2.2. How does the censoring rate affect the power of the test? 19112.4. Statistical inference 19112.5. Acknowledgment 19212.6. Bibliography 193Chapter 13. Some Asymptotic Results for Exchangeably Weighted Bootstraps of the Empirical Estimator of a Semi-Markov Kernel with Applications 195Salim BOUZEBDA and Nikolaos LIMNIOS13.1. Introduction 19513.2. Semi-Markov setting 19713.3. Main results 20113.4. Bootstrap for a multidimensional empirical estimator of a continuous-time semi-Markov kernel 20513.5. Confidence intervals 20813.6. Bibliography 210Chapter 14. On Chi-Squared Goodness-of-Fit Test for Normality 213Mikhail NIKULIN, Léo GERVILLE-RÉACHE and Xuan Quang TRAN14.1. Chi–squared test for normality 21314.2. Simulation study 22114.3. Bibliography 226Part 2. Statistical Models and Methods in Survival Analysis 229Chapter 15. Estimation/Imputation Strategies for Missing Data in Survival Analysis 231Elodie BRUNEL, Fabienne COMTE and Agathe GUILLOUX15.1. Introduction 23115.2. Model and strategies 23315.2.1. Model assumptions 23315.2.2. Strategy involving knowledge of ζ 23415.2.3. Strategy involving knowledge of π 23515.2.4. Estimation of ζ or π: logit or non-parametric regression 23615.2.5. Computing the hazard estimators 23615.2.6. Theoretical results 23915.3. Imputation-based strategy 24115.4. Numerical comparison 24215.5. Proofs 24415.6. Bibliography 251Chapter 16. Non-Parametric Estimation of Linear Functionals of a Multivariate Distribution Under Multivariate Censoring with Applications 253Olivier LOPEZ and Philippe SAINT-PIERRE16.1. Introduction 25316.2. Non-parametric estimation of the distribution 25516.3. Asymptotic properties 25716.4. Statistical applications of functionals 26016.4.1. Dependence measures 26016.4.2. Bootstrap 26116.4.3. Linear regression 26216.5. Illustration 26316.6. Conclusion 26416.7. Acknowledgment 26416.8. Bibliography 264Chapter 17. Kernel Estimation of Density from Indirect Observation 267Valentin SOLEV17.1. Introduction 26717.1.1. Random partition 26717.1.2. Indirect observation 26817.1.3. Kernel density estimator 26917.2. Density of random vector Λ(X) 27117.3. Pseudo-kernel density estimator 27317.3.1. Pointwise density estimation based on indirect data 27317.3.2. Bias of the kernel estimator 27417.3.3. Estimate of variance 27617.4. Bibliography 279Chapter 18. A Comparative Analysis of Some Chi-Square Goodness-of-Fit Tests for Censored Data 281Ekaterina CHIMITOVA and Boris LEMESHKO18.1. Introduction 28118.2. Chi-square goodness-of-fit tests for censored data 28318.2.1. NRR χ2 test 28318.2.2. GPF χ2 test 28418.3. The choice of grouping intervals 28518.3.1. Equifrequent grouping (EFG) 28918.3.2. Intervals with equal expected numbers of failures (EENFG) 28918.3.3. Optimal grouping (OptG) 28918.4. Empirical power study 29018.5. Conclusions 29318.6. Acknowledgment 29418.7. Bibliography 294Chapter 19. A Non-parametric Test for Comparing Treatments with Missing Data and Dependent Censoring 297Amel MEZAOUER, Kamal BOUKHETALA and Jean-François DUPUY19.1. Introduction 29719.2. The proposed test statistic 29919.3. Asymptotic distribution of the proposed test statistic 30119.4. Acknowledgment 30519.5. Appendix 30619.6. Bibliography 309Chapter 20. Group Sequential Tests for Treatment Effect with Covariates Adjustment through Simple Cross-Effect Models 311Isaac Wu HONG-DAR20.1. Introduction 31120.2. Notations and models 31320.3. Group sequential test 31620.4. Discussion 31820.5. Acknowledgment 31820.6. Bibliography 318Part 3. Reliability and Maintenance 321Chapter 21. Optimal Maintenance in Degradation Processes 323Waltraud KAHLE21.1. Introduction 32321.2. The degradation model 32421.3. Optimal replacement after an inspection 32621.4. The simulation of degradation processes 32721.5. Shape of cost functions and optimal δ and a 32921.6. Incomplete preventive maintenance 33021.7. Bibliography 333Chapter 22. Planning Accelerated Destructive Degradation Tests with Competing Risks 335Ying SHI and William Q. MEEKER22.1. Introduction 33622.1.1. Background 33622.1.2. Motivation: adhesive bond C 33622.1.3. Related literature 33722.1.4. Overview 33822.2. Degradation models with competing risks 33822.2.1. Accelerated degradation model for the primary response 33822.2.2. Accelerated degradation model for the competing response 33922.2.3. Degradation models for adhesive bond C 33922.2.4. Degradation distribution and quantiles 34022.3. Failure-time distribution with competing risks 34122.3.1. Relationship between degradation and failure 34122.3.2. Failure-time distribution and quantiles 34222.4. Test planning with competing risks 34222.4.1. ADDT planning information 34222.4.2. Criterion for ADDT planning with competing risks 34322.5. ADDT plans with competing risks 34422.5.1. Initial optimum ADDT plan with competing risks 34422.5.2. Constrained optimum ADDT plan with competing risks 34822.5.3. General equivalence theorem 34822.5.4. Compromise ADDT plan with competing risks 35022.6. Monte Carlo simulation to evaluate test plans 35222.7. Conclusions and extensions 35322.8. Appendix: technical details 35422.8.1. The Fisher information matrix for ADDT with competing risks 35422.8.2. Large-sample approximate variance of ht (tp) and tp 35522.9. Bibliography 355Chapter 23. A New Goodness-of-Fit Test for Shape-Scale Families 357Vilijandas BAGDONAVIČIUS23.1. Introduction 35723.2. The test statistic 35823.3. The asymptotic distribution of the test statistic 35923.4. The test 36423.5. Weibull distribution 36423.6. Loglogistic distribution 36523.7. Lognormal distribution 36623.8. Bibliography 367Chapter 24. Time-to-Failure of Markov-Modulated Gamma Process with Application to Replacement Policies 369Christian PAROISSIN and Landy RABEHASAINA24.1. Introduction 36924.2. Degradation model 37024.2.1. Covariate process 37024.2.2. Degradation process 37124.3. Time-to-failure distribution 37124.3.1. Case of a non-modulated gamma process 37224.3.2. Case of a Markov-modulated gamma process 37324.3.3. Stochastic comparison 37424.4. Replacement policies 37624.4.1. Block replacement policy 37724.4.2. Age replacement policy 37924.5. Conclusion 38124.6. Acknowledgment 38124.7. Bibliography 382Chapter 25. Calculation of the Redundant Structure Reliability for Agingtype Elements 383Alexandr ANTONOV, Alexandr PLYASKIN and Khizri TATAEV25.1. Introduction 38325.2. The operation process of the renewal and repaired products 38425.3. The model of the geometric process 38625.4. Task solution 38725.5. Conclusion 38925.6. Bibliography 390Chapter 26. On Engineering Risks of Complex Hierarchical Systems Analysis 391Vladimir RYKOV26.1. Introduction 39126.2. Risk definition and measurement 39226.3. Engineering risk 39326.4. Risk characteristics for general model calculation 39526.4.1. Lifelength and appropriate loss size CDF 39526.4.2. Probability of risk event evolution 39626.4.3. Lifelength and loss moments 39726.4.4. Mostly dangerous paths of risk event evolution and sensitivity analysis 39926.5. Risk analysis for short-time risk models 40026.6. Conclusion 40226.7. Bibliography 402List of Authors 405Index 409