Advances in Degradation Modeling: Applications to Reliability, Survival Analysis, and Finance (Statistics for Industry and Technology)

This volume—dedicated to William Q. Meeker on the occasion of his sixtieth birthday—is a collection of invited chapters covering recent advances in accelerated life testing and degradation models. The book covers a wide range of applications to areas such as reliability, quality control, the health sciences, economics, and finance. Specific topics covered include: \* Accelerated testing and inference \* Step-stress testing and inference \* Nonparametric inference \* Model validity in accelerated testing \* The point process approach \* Bootstrap methods in degradation analysis \* Exact inferential methods in reliability \* Dynamic perturbed systems \* Degradation models in statistics __Advances in Degradation Modeling__ is an excellent reference for researchers and practitioners in applied probability and statistics, industrial statistics, the health sciences, quality control, economics, and finance. Contents 6 Preface 15 William Q. Meeker -- Career and Accomplishments 17 List of Contributors 26 List of Tables 30 List of Figures 32 Part I Review, Tutorials, and Perspective 36 1 Trends in the Statistical Assessment of Reliability 37 William Q. Meeker 37 1.1 Background and Purpose 37 1.2 Traditional Reliability Data and Data Analysis 38 1.3 Product Design and Reliability Budgeting 39 1.4 Accelerated Testing 40 1.5 Multiple Failure Modes 40 1.6 Field and Warranty Data 42 1.7 Degradation Reliability Data 42 1.8 Recurrence Data 43 1.9 The Next Generation of Reliability Data 44 1.10 Software for Statistical Analysis of Reliability Data 45 1.11 Use of Bayesian Methods in Reliability 46 1.12 Concluding Remarks 48 References 48 2 Degradation Processes: An Overview 51 Mohamed Abdel-Hameed 51 2.1 Introduction 51 2.2 Lévy and Pure Jump Processes 52 2.2.1 Lévy processes 52 2.2.2 Pure jump Markov processes 54 2.3 Life Distributions and Dependence Between Random Variables 54 2.3.1 Classes of life distributions 54 2.3.2 Dependence between random variables 55 2.4 The Degradation Process and Properties of the Resulting Life Distribution 55 2.4.1 Non-stationary gamma degradation process 55 2.4.2 Increasing Lévy and pure jump degradation processes 56 2.4.3 Brownian motion like degradation processes 56 2.5 Maintenance Policies of Devices Subject to Degradation 57 References 58 3 Defect Initiation, Growth, and Failure -- A General Statistical Model and Data Analyses 60 Wayne B. Nelson 60 3.1 Introduction 60 3.2 An Application 61 3.3 A Population Model for Defect Initiation and Growth 63 3.4 Model Fitting by Computer 67 References 70 4 Properties of Lifetime Estimators Basedon Warranty Data Consisting only of Failures 71 Kazuyuki Suzuki, Watalu Yamamoto, Takashi Hara, and Md. Mesbahul Alam 71 4.1 Introduction 71 4.2 A Model for Failure Data in Warranty Database 73 4.3 Maximum Likelihood Estimation 74 4.4 Properties of Maximum Likelihood Estimators 76 4.4.1 Fisher information matrix 76 4.4.2 One failure mode case 76 4.4.3 Two failure modes case 76 4.5 Effect of Sales Amount on Precision 78 4.6 Conclusion 80 A Gradients and Hessians of Log-Likelihood (4.3) 80 A.1 Gradients 81 A.2 Hessians 81 B Expectation of the Elements of Hessians 83 C Gauss--Hermite Quadrature 86 References 87 Part II Shock Models 88 5 Shock Models 89 Allan Gut and Jürg Hüsler 89 5.1 Introduction 89 5.2 Cumulative Shock Models 90 5.3 Extreme Shock Models 90 5.4 Mixed Shock Models 91 5.4.1 An auxiliary result 92 5.4.2 Some examples 93 5.5 More Realistic Model 93 5.5.1 Delayed sums 94 5.5.2 A generalized cumulative shock model 95 5.5.3 A generalized extreme shock model 96 5.5.4 A generalized mixed shock model 98 5.5.5 Extreme shock models with varying threshold 99 5.5.6 The exact distribution of 100 5.5.7 The asymptotic distribution 101 5.6 An Extension to Markovian Shock Models 103 5.6.1 Cumulative shock models 104 5.6.2 Extreme shock models 104 5.6.3 Mixed shock models 105 References 105 6 Parametric Shock Models 107 Waltraud Kahle and Heide Wendt 107 6.1 Introduction 107 6.2 Modeling Degradation by Marked Point Processes 108 6.3 Characteristics of the Degradation Process 112 6.3.1 The counting process ((t)) 112 6.3.2 The sequence (Xn) and the degradation process (Zt) 113 6.3.3 The cumulative degradation at time t 114 6.3.4 The first passage time Zh 118 6.4 Maximum Likelihood Estimations 120 6.5 The Large Sample Case 127 6.6 Moment Estimates 129 6.7 Comparison of Maximum Likelihood and Moment Estimates 131 6.8 Conclusion 133 References 133 7 Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching -- Toward Application in Reliability 135 V.S. Koroliuk, N. Limnios, and I.V. Samoilenko 135 7.1 Introduction 135 7.2 Main Results 137 7.3 Proof of Theorem 1 140 References 145 8 On Some Shock Models of Degradation 147 Maxim Finkelstein and Ji Hwan Cha 147 8.1 Introduction 147 8.2 Asymptotic Properties for Two Special Shock Models 149 8.3 Terminating and ``Accumulating'' Shocks 151 8.4 Concluding Remarks 154 References 154 Part III Degradation Models 155 9 The Wiener Process as a Degradation Model: Modeling and Parameter Estimation 156 Waltraud Kahle and Axel Lehmann 156 9.1 Introduction 156 9.2 Increments of the Degradation Process Are Observed 158 9.3 Observation of Failure Times 163 9.4 Observation of Both Degradation Increments and Failure Times 164 9.5 An Example 170 9.6 Simulation Study 173 References 174 10 On the General Degradation Path Model: Review and Simulation 176 Firoozeh Haghighi, Nazanin Nooraee, and Narges Nazeri Rad 176 10.1 Introduction 176 10.2 Degradation Model 177 10.2.1 Degradation model with noise 178 10.2.2 Degradation model without measurement error 179 10.3 Simulation 180 References 183 11 A Closer Look at Degradation Models: Classicaland Bayesian Approaches 185 Marta A. Freitas, Thiago R. dos Santos, Magda C. Pires, and Enrico A. Colosimo 185 11.1 Introduction 185 11.1.1 Background and literature 185 11.1.2 The problem 187 11.2 Train Wheel Degradation Data 189 11.3 Statistical Methods for Degradation Data Analysis 190 11.3.1 Methods based on ``classical" inference 190 The approximate method 190 The analytical method 191 The two-stage method 192 The numerical method 193 11.3.2 Bayesian inference 194 11.4 The Wheel Degradation Data Revisited 195 11.4.1 Estimation of FT(t) using the approximate method 196 11.4.2 Estimation of FT(t) using the analytical method 198 11.4.3 Estimation of FT(t) using the two-stage method 199 11.4.4 Estimation of FT(t) using the numerical method 200 11.4.5 Comparison of the results generated by the methods based on ``classical" inference 200 11.4.6 Bayesian inference 200 11.5 Conclusions 205 References 206 12 Optimal Prophylaxis Policy Under Non-monotone Degradation 209 S.S. Rasova and B.P. Harlamov 209 12.1 Setting of the Problem 209 12.2 Optimization Problem 211 12.3 Trajectories of Choice 212 12.4 Degradation Process of Diffusion Type 220 References 222 13 Deterioration Processes with Increasing Thresholds 223 S. Zacks 223 13.1 Introduction 223 13.2 Preliminaries 224 13.3 The Reliability and Hazard in Case I 225 13.4 The Reliability and Hazard in Case II 227 13.5 Reliability and Hazard in Case III 228 13.6 Reliability and Hazard in Case IV 229 13.7 Exponential Deterioration 230 13.7.1 Case I with K=1 230 13.7.2 Case I with K=2 231 13.7.3 Case II with K=1 232 13.7.4 Case III, K = 1 234 13.7.5 Case IV: Linear threshold 235 References 236 14 Failure Time Models Based on Degradation Processes 237 Axel Lehmann 237 14.1 Introduction 237 14.1.1 Models for degradation data 238 14.1.2 Models relating degradation and failure 241 14.2 Degradation--Threshold--Shock Models 242 14.2.1 The general model 242 14.2.2 Degradation--threshold models 244 14.2.3 Degradation--shock models 245 14.2.4 Likelihood function 246 14.2.5 Estimation of the survival function and the failure rate of T 250 14.3 DTS Models with Covariates 250 14.3.1 Maximum likelihood estimation 253 14.3.2 Semiparametric estimation 255 14.4 A DTS Model for Repairable Items 257 14.5 Application of DTS Models 258 References 259 15 Degradation and Fuzzy Information 262 R. Viertl 262 15.1 Introduction 262 15.2 Material Degradation 263 15.3 Fuzzy Initial Conditions 264 15.4 Fuzzy Distribution of q(0) 265 15.5 Application to Accelerated Life Testing 267 References 267 16 A New Perspective on Damage Accumulation, Marker Processes, and Weibull's Distribution 268 Nozer D. Singpurwalla 268 16.1 The Hazard Potential 269 16.2 A Stochastic Process Model for Damage and Its Markers 270 16.3 Introduction 271 16.4 The Weibull Distribution in Material Failure: Some History 271 16.5 Preliminaries and Notation 271 16.6 The Weakest Link Principle and an Application 272 16.7 Weibull's Approximation and Analysis 273 16.8 Critique of Weibull's Analysis 274 16.9 The Theory of Extreme Values 275 16.10 Comments on Using the Weibull Distribution Motivated via WLP and EVT 275 References 276 Part IV Reliability Estimation and ALT 277 17 Reliability Estimation of Mechanical Components Using Accelerated Life Testing Models 278 Fabrice Guérin, M. Barreau, A. Charki, A. Todoskoff, S. Cloupet and D. Bigaud 278 17.1 Introduction 278 17.2 Accelerated Life Testing Model 279 17.3 Regression Test Plan 281 17.3.1 Introduction 281 17.3.2 Parametric ALT model 282 17.3.3 Generalized proportional hazard (GPH) model 282 17.3.4 Semi-parametric ALT model 284 17.3.5 Application to ball bearings 285 Ball bearing life 285 Simulation model definition of testing result 285 Testing result analysis 286 Asymptotic behavior study 286 Analysis 289 17.4 Reliability Test With Previous Accelerated Damage 291 17.4.1 Principle 291 17.4.2 Test plan definition 292 17.4.3 Parametric model 293 17.4.4 Nonparametric model 294 17.4.5 Simulation example 295 17.5 Conclusions 297 References 298 18 Reliability Estimation from Failure-Degradation Data with Covariates 300 V. Bagdonavicius, I. Masiulaityte, M.S. Nikulin 300 18.1 Introduction 300 18.2 Modelling Simultaneous Traumatic Events and Degradation Data Under Covariates 301 18.3 Estimation of Model Parameters 305 18.3.1 The data 305 18.3.2 Likelihood function construction 305 18.3.3 Example 1: Time-scaled gamma process 306 (a) Parametric form of the mean degradation 307 (b) Unknown form of mean degradation 308 18.3.4 Example 2: Shock processes 309 (a) Parametric form of the mean degradation 310 (b) Unknown form of mean degradation 311 18.3.5 Example 3: Path models 311 18.3.6 Modified loglikelihood 313 18.4 Estimation of Reliability Characteristics 313 References 314 19 Asymptotic Properties of Redundant Systems Reliability Estimators 317 V. Bagdonavicius, I. Masiulaityte, M.S. Nikulin 317 19.1 Introduction 317 19.2 Point Estimators of the c.d.f. of Redundant Systems 318 19.2.1 Nonparametric estimation 318 19.2.2 Parametric estimation 320 19.3 Asymptotic Distribution of j and Confidence Intervals for Kj(t) 320 19.3.1 Nonparametric case 321 19.3.2 Parametric case 326 19.4 Power of Goodness-of-Fit Tests 330 References 334 20 An Approach to System Reliability Demonstration Based on Accelerated Test Results on Components 335 Léo Gerville-Réache and Vincent Couallier 335 20.1 Introduction 335 20.2 Global Reliability Demonstration from k ``Zero-Failure'' Component Testing Procedures 337 20.2.1 Equal component test times 338 20.2.2 Equal reliability targets 338 20.2.3 Integrating failed demonstration procedures 339 20.3 Designing a Global Demonstration Test for the Reliability of a Series System Under a Success Probability Constraint 340 20.3.1 Basic principle of the demonstration test planning 340 20.3.2 ``n/j failures'' demonstration test 342 20.4 Conclusion 343 References 343 Part V Survival Function Estimation 345 21 Robust Versus Nonparametric Approachesand Survival Data Analysis 346 Catherine Huber 346 21.1 Introduction 347 21.2 Motivation for Robustness 347 21.2.1 Instability of usual tests and estimators 347 21.2.2 Sensitivity of rank test 350 21.3 Robustness Concepts 351 21.3.1 Robust versus nonparametric approach 351 21.3.2 Regularity of the parametric model 352 21.3.3 Extension of the underlying model 353 21.3.4 Measures of robustness 353 21.4 Robustness in Survival Analysis 354 21.4.1 Measure of robustness of Kaplan--Meier estimator 354 Peterson representation of Kaplan--Meier estimator 355 21.4.2 Perspectives 359 References 359 22 Modelling Recurrent Events for Repairable Systems Under Worse Than Old Assumption 361 G. Babykina and V. Couallier 361 22.1 Introduction 361 22.2 A New Model of Imperfect Repair: The LEYP Model 364 22.2.1 Some useful properties 366 22.3 Taking Covariates into Account 367 22.4 Statistical Estimation and Data Description 367 22.5 Numerical Example 369 22.5.1 Data description 369 22.5.2 Parameter estimation 372 22.5.3 Predictions 374 22.6 Conclusion 375 References 375 23 Survival Models for Step-Stress Experiments With Lagged Effects 377 N. Kannan, D. Kundu, and N. Balakrishnan 377 23.1 Introduction 377 23.2 Model Description 379 23.2.1 Step-stress models with latency 379 23.3 Maximum Likelihood Estimators for the CRM 380 23.4 Least Squares Estimators 382 23.5 Data Analysis 383 23.6 Simulation Results 384 23.7 Conclusions 389 References 390 24 Estimation of Density on Censored Data 392 V. Solev 392 24.1 Introduction 392 24.2 Approximating of Parametric Set 395 24.3 Hellinger Distance 397 24.4 Main Result 399 References 400 Part VI Competing Risk and Chaotic Systems 401 25 Toward a Test for Departure of a Trajectory from a Neighborhood of a Chaotic System 402 M. LuValle 402 25.1 Introduction 402 25.1.1 Terminology and the Lorenz attractor 404 25.1.2 The alternative for the simulation 405 25.2 The Test Statistic and Supporting Theory 406 25.3 Computer Experiments 410 25.3.1 Description of the computer experiments 411 25.4 Directions for Future Work 413 References 413 26 Probability Plotting with Independent Competing Risks 416 Francis G. Pascual and Christopher Gast 416 26.1 Introduction 416 26.1.1 Competing risks 416 26.1.2 Probability plotting 417 26.1.3 Outline 418 26.2 Notation and Model Assumptions 418 26.2.1 Distributions of individual risks 418 26.2.2 Distribution of subject lifetime 419 26.2.3 The likelihood function 419 26.3 The Kaplan--Meier Estimator And Probability Plotting 420 26.3.1 Kaplan--Meier estimator 420 26.3.2 Linearizing the Cdf under one risk 421 26.3.3 Probability plotting and competing risks 421 26.4 Proposed Method 422 26.5 Applications 423 26.5.1 Breast cancer study 423 26.5.2 Shock absorber failure data 426 26.5.3 Simulated data set 429 26.6 Conclusions 431 References 432 Index 434 Front Matter....Pages i-xxxviii Front Matter....Pages 1-1 Trends in the Statistical Assessment of Reliability....Pages 3-16 Degradation Processes: An Overview....Pages 17-25 Defect Initiation, Growth, and Failure – A General Statistical Model and Data Analyses....Pages 27-37 Properties of Lifetime Estimators Based on Warranty Data Consisting only of Failures....Pages 39-55 Front Matter....Pages 56-56 Shock Models....Pages 59-76 Parametric Shock Models....Pages 77-104 Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching – Toward Application in Reliability....Pages 105-116 On Some Shock Models of Degradation....Pages 117-124 Front Matter....Pages 125-125 The Wiener Process as a Degradation Model: Modeling and Parameter Estimation....Pages 127-146 On the General Degradation Path Model: Review and Simulation....Pages 147-155 A Closer Look at Degradation Models: Classical and Bayesian Approaches....Pages 157-180 Optimal Prophylaxis Policy Under Non-monotone Degradation....Pages 181-194 Deterioration Processes With Increasing Thresholds....Pages 195-208 Failure Time Models Based on Degradation Processes....Pages 209-233 Degradation and Fuzzy Information....Pages 235-240 A New Perspective on Damage Accumulation, Marker Processes, and Weibull’s Distribution....Pages 241-249 Front Matter....Pages 250-250 Reliability Estimation of Mechanical Components Using Accelerated Life Testing Models....Pages 253-274 Reliability Estimation from Failure-Degradation Data with Covariates....Pages 275-291 Asymptotic Properties of Redundant Systems Reliability Estimators....Pages 293-310 An Approach to System Reliability Demonstration Based on Accelerated Test Results on Components....Pages 311-320 Front Matter....Pages 321-321 Robust Versus Nonparametric Approaches and Survival Data Analysis....Pages 323-337 Modelling Recurrent Events for Repairable Systems Under Worse Than Old Assumption....Pages 339-354 Survival Models for Step-Stress Experiments With Lagged Effects....Pages 355-369 Estimation of Density on Censored Data....Pages 371-379 Front Matter....Pages 380-380 Toward a Test for Departure of a Trajectory from a Neighborhood of a Chaotic System....Pages 383-396 Probability Plotting with Independent Competing Risks....Pages 397-414 Back Matter....Pages 415-416 William Q. Meeker has made pioneering and phenomenal contributions to the general areaofreliabilityand,inparticular,tothetopicsofdegradationandacceleratedtesting. Hisresearchpublicationsandthenumerouscitationshehasreceivedoverthepastthree decades provide an ample testimony to this fact. Statistical methods have become critical in analyzing reliability and survival data. Highly reliable products have necessitated the development of accelerated testing and degradation models and their analyses. This volume has been put together in order to (i) review some of the recent advances on accelerated testing and degradation, (ii) highlight some new results and discuss their applications, and (iii) suggest possible directions for future research in these topics. With these speci?c goals in mind, many authors were invited to write a chapter for this volume. These authors are not only experts in lifetime data analysis, but also form a representative group from former students, colleagues, and other close professional associates of William Meeker. All contributions have been peer reviewed and organized into 26 chapters. For the convenience of readers, the volume has been divided into the following six parts: • Review, Tutorials, and Perspective • Shock Models • Degradation Models • Reliability Estimation and ALT • Survival Function Estimation • Competing Risk and Chaotic Systems Itneedstobeemphasizedherethatthisvolumeisnotaproceedings,butacarefully anddeliberatelyplannedvolumecomprisingchaptersconsistentwiththeeditorialgoals and purposes mentioned above. Our thanks go to all the authors who have contributed to this volume. Thanks are also due to Mrs. Debbie Iscoe for the excellent typesetting of the entire volume.SpecialthanksgotoMs.ReginaGorenshteynandMr.TomGrasso(Editor,Birkh ̈ auser, Boston) for their interest and support for this project.