This book improves Meeker and Escobar (1998, Wiley) not only in terms of organization and presentation, but also in extensions and modifications to the technical material. It presents state-of-the-art, computer-based statistical methods for reliability data analysis and for test planning of industrial products. It also improves long time established techniques as it demonstrates how to apply the new graphical, numerical, or simulation-based methods to a broad range of models encountered in reliability data analysis. Bayesian methods in solving practical problems (e.g. models involving random effects or censoring that arises in reliability studies) are now incorporated where appropriate; the computations are done with WinBUGS. Ample exercises that extend and strengthen the concepts in the book are included. The criterion for integrating material in the book is that the authors have in-hand or have seen real applications for the methodology. The book is specifically geared for a one-semester course on the topic in either a statistics or engineering department at either the upper-undergraduate or beginning graduate levels. R functions and subroutines, along with an extensive list of data sets, are included on a massive web site that is meticulously maintained by the authors.

William Q. Meeker, PhD, is Professor of Statistics and Distinguished Professor of Liberal Arts and Sciences at Iowa State University. He is a Fellow of the American Association for the Advancement of Science, the American Statistical Association, and the American Society for Quality.

Luis A. Escobar, PhD, is a Professor in the Department of Experimental Statistics at Louisiana State University. He is a Fellow of the American Statistical Association, an elected member of the International Statistics Institute, and an elected Member of the Colombian Academy of Sciences.

Francis G. Pascual, PhD, is an Associate Professor in the Department of Mathematics and Statistics at Washington State University.

Statistical Methods for Reliability Data i

Preface to the Second Edition iii

Preface to First Edition viii

Acknowledgments xii

1 Reliability Concepts and an Introduction to Reliability Data 1

1.1 Introduction 1

1.1.1 Quality and Reliability 1

1.1.2 Reasons for Collecting Reliability Data 2

1.1.3 Distinguishing Features of Reliability Data 2

1.2 Examples of Reliability Data 3

1.2.1 Failure-Time Data with No Explanatory Variables 3

1.2.2 Failure-Time Data with Explanatory Variables 8

1.2.3 Degradation Data with No Explanatory Variables 10

1.2.4 Degradation Data with Explanatory Variables 11

1.3 General Models for Reliability Data 11

1.3.1 Reliability Studies and Processes 11

1.3.2 Causes of Failure and Degradation Leading to Failure 11

1.3.3 Environmental Effects on Reliability 12

1.3.4 Definition of Time Scale 12

1.3.5 Definitions of Time Origin and Failure Time 13

1.4 Models for Time to Event Versus Models for Recurrences in a Sequence of Events 13

1.4.1 Modeling Times to an Event 14

1.4.2 Modeling a Sequence of Recurrent Events 14

1.5 Strategy for Data Collection, Modeling, and Analysis 15

1.5.1 Planning a Reliability Study 15

1.5.2 Strategy for Data Analysis and Modeling 15

2 Models, Censoring, and Likelihood for Failure-Time Data 19

2.1 Models for Continuous Failure-Time Processes 19

2.1.1 Failure-Time Probability Distribution Functions 20

2.1.2 The Quantile Function and Distribution Quantiles 22

2.1.3 Distribution of Remaining Life 23

2.2 Models for Discrete Data from a Continuous Process 25

2.2.1 Multinomial Failure-Time Model 25

2.2.2 Multinomial cdf 26

2.3 Censoring 27

2.3.1 Censoring Mechanisms 27

2.3.2 Important Assumptions on Censoring Mechanisms 28

2.3.3 Informative Censoring 28

2.4 Likelihood 28

2.4.1 Likelihood-Based Statistical Methods 28

2.4.2 Specifying the Likelihood Function 28

2.4.3 Contributions to the Likelihood Function 29

2.4.4 Form of the Constant Term C 31

2.4.5 Likelihood Terms for General Reliability Data 32

2.4.6 Other Likelihood Terms 32

3 Nonparametric Estimation for Failure-Time Data 37

3.1 Estimation from Complete Data 38

3.2 Estimation from Singly-Censored Interval Data 38

3.3 Basic Ideas of Statistical Inference 40

3.3.1 The Sampling Distribution of b F(ti) 40

3.3.2 Confidence Intervals 41

3.4 Confidence Intervals from Complete or Singly-Censored Data 41

3.4.1 Pointwise Binomial-Based Conservative Confidence Interval for F(ti) 41

3.4.2 Pointwise Binomial-Based Jeffreys Approximate Confidence Interval for F(ti) 42

3.4.3 Pointwise Wald Approximate Confidence Interval for F(ti) 42

3.5 Estimation from Multiply-Censored Data 43

3.6 Pointwise Confidence Intervals from Multiply-Censored Data 45

3.6.1 Approximate Variance of b F(ti) 45

3.6.2 Greenwood’s Formula 46

3.6.3 Pointwise Wald Confidence Interval for F(ti) 46

3.7 Estimation from Multiply-Censored Data with Exact Failures 47

3.8 Nonparametric Simultaneous Confidence Bands 49

3.8.1 Motivation 49

3.8.2 Nonparametric Simultaneous Large-Sample Approximate Confidence Bands for F(t) 50

3.8.3 Determining the Time Range for Nonparametric Simultaneous Confidence Bands for F(t) 51

3.9 Arbitrary Censoring 52

4 Some Parametric Distributions Used in Reliability Applications 60

4.1 Introduction 61

4.2 Quantities of Interest in Reliability Applications 61

4.3 Location-Scale and Log-Location-Scale Distributions 62

4.4 Exponential Distribution 63

4.4.1 CDF, PDF, Moments, HF, and Quantile Functions 63

4.4.2 Motivation and Applications 63

4.5 Normal Distribution 64

4.5.1 CDF, PDF, Moments, and Quantile Function 64

4.5.2 Motivation and Applications 64

4.6 Lognormal Distribution 65

4.6.1 CDF, PDF, Moments, and Quantile Function 65

4.6.2 Motivation and Applications 66

4.7 Smallest Extreme Value Distribution 67

4.7.1 CDF, PDF, Moments, HF, and Quantile Functions 67

4.7.2 Motivation and Applications 67

4.8 Weibull Distribution 68

4.8.1 CDF, Moments, and Quantile Function 68

4.8.2 Alternative Parameterization 68

4.8.3 Alternative Parameterization CDF, PDF, HF, and Quantile Function 69

4.8.4 Motivation and Applications 69

4.9 Largest Extreme Value Distribution 70

4.9.1 CDF, PDF, Moments, HF, and Quantile Function 70

4.9.2 Motivation and Applications 71

4.10 Fr´echet Distribution 71

4.10.1 CDF, Moments, and Quantile Function 71

4.10.2 Alternative Parameterization 71

4.10.3 CDF, PDF, and Quantile Function in the Alternative Parameterization 72

4.10.4 Motivation and Applications 72

4.11 Logistic Distribution 73

4.11.1 CDF, PDF, Moments, and Quantile Function 73

4.11.2 Similarity with the Normal Distribution 73

4.12 Loglogistic Distribution 74

4.12.1 CDF and PDF 74

4.12.2 Moments and Quantile Function 74

4.12.3 Motivation and Applications 75

4.13 Generalized Gamma Distribution 75

4.13.1 CDF and PDF 75

4.13.2 Moments and Quantile Function 76

4.13.3 Special Cases of the Generalized Gamma Distribution 76

4.14 Distributions with a Threshold Parameter 76

4.15 Other Methods of Deriving Failure-Time Distributions 78

4.15.1 Discrete Mixture Distributions 78

4.15.2 Continuous Mixture Distributions 78

4.15.3 Power Distributions 79

4.16 Parameters and Parameterization 80

4.17 Generating Pseudorandom Observations from a Specified Distribution 80

4.17.1 Uniform Pseudorandom Number Generator 80

4.17.2 Pseudorandom Observations from Continuous Distributions 80

4.17.3 Efficient Generation of Pseudorandom Censored Samples 80

4.17.4 Pseudorandom Observations from Discrete Distributions 82

5 System Reliability Concepts and Methods 87

5.1 Non-Repairable System Reliability Metrics 88

5.1.1 System cdf 88

5.1.2 Other Non-Repairable System Reliability Metrics 88

5.2 Series Systems 88

5.2.1 Probability of Failure for a Series System Having Components with Independent Failure Times 88

5.2.2 Importance of Part Count in Product Design 89

5.2.3 Series System of Independent Components Having Weibull Distributions with the Same Shape Parameter 90

5.2.4 Effect of Positive Dependency in a Two-Component Series System 90

5.3 Parallel Systems 91

5.3.1 The Effect of Parallel Redundancy in Improving (Sub)-System Reliability 91

5.3.2 Effect of Positive Dependency in a Two-Component Parallel-Redundant System 92

5.3.3 Another Kind of Redundancy 92

5.4 Series-Parallel Systems 93

5.4.1 Series-Parallel Systems with System-Level Redundancy 93

5.4.2 Series-Parallel System Structure with Component-Level Redundancy 94

5.5 Other System Structures 94

5.5.1 Bridge System Structures 94

5.5.2 k-out-of-m System Structure 95

5.5.3 k-out-of-m: F (failed) systems 95

5.6 Multistate System Reliability Models 96

5.6.1 Nonrepairable Multistate Systems 96

5.6.2 Repairable Multistate Systems 96

5.6.3 Repairable System Availability 97

5.6.4 Repairable System and Mean Time Between Failures 97

6 Probability Plotting 102

6.1 Introduction 103

6.2 Linearizing Location-Scale-Based Distributions 103

6.2.1 Linearizing the Exponential Distribution cdf 103

6.2.2 Linearizing the Normal Distribution cdf 103

6.2.3 Linearizing the Lognormal Distribution cdf 104

6.2.4 Linearizing the Weibull Distribution cdf 104

6.2.5 Linearizing the cdf of Other Location-Scale or Log-Location-Scale Distributions 105

6.3 Graphical Goodness of Fit 105

6.4 Probability Plotting Positions 106

6.4.1 Criteria for Choosing Plotting Positions 106

6.4.2 Choice of Plotting Positions 106

6.4.3 Summary of Probability Plotting Methods 111

6.5 Notes on the Application of Probability Plotting 111

6.5.1 Using Simulation to Help Interpret Probability Plots 111

6.5.2 Possible Reason for a Bend in a Probability Plot 114

7 Parametric Likelihood Fitting Concepts: Exponential Distribution 119

7.1 Introduction 120

7.1.1 Maximum Likelihood Background 120

7.1.2 Model Selection 121

7.2 Parametric Likelihood 122

7.2.1 Probability of the Data 122

7.2.2 Likelihood Function and its Maximum 122

7.3 Likelihood Confidence Intervals for θ 123

7.3.1 Confidence Intervals Based on a Profile Likelihood 123

7.3.2 Relationship Between Confidence Intervals and Significance Tests 124

7.4 Wald (Normal-Approximation) Confidence Intervals for θ 125

7.5 Confidence Intervals for Functions of θ 126

7.5.1 Confidence Intervals for the Arrival Rate 127

7.5.2 Confidence Intervals for F(t; θ) 127

7.6 Comparison of Confidence Interval Procedures 127

7.7 Likelihood for Exact Failure Times 128

7.7.1 Correct Likelihood for Observations Reported as Exact Failures 128

7.7.2 Using the Density Approximation for Observations Reported as Exact Failures 128

7.7.3 ML Estimates for the Exponential Distribution θ Based on the Density Approximation 128

7.7.4 Confidence Intervals for the Exponential Distribution with Complete Data or Type 2 (Failure) Censoring 129

7.8 Effect of Sample Size on Confidence Interval Width and the Likelihood Shape 130

7.8.1 Effect of Sample Size Confidence Interval Width 130

7.8.2 Effect of Sample Size on the Likelihood Shape 130

7.9 Exponential Distribution Inferences with No Failures 131

8 Maximum Likelihood Estimation for Log-Location-Scale Distributions 138

8.1 Likelihood Definition 139

8.1.1 The Likelihood for Location-Scale Distributions 139

8.1.2 The Likelihood for Log-Location-Scale Distributions 139

8.1.3 Akaike Information Criterion 141

8.2 Likelihood Confidence Regions and Intervals 142

8.2.1 Joint Confidence Regions for μ and σ 142

8.2.2 Likelihood Confidence Intervals for μ 142

8.2.3 Likelihood Confidence Intervals for σ 143

8.2.4 Likelihood Confidence Intervals for Functions of μ and σ 143

8.2.5 Relationship between Confidence Intervals and Significance Tests 145

8.3 Wald Confidence Intervals 146

8.3.1 Variance-Covariance Matrix of Parameter Estimates 146

8.3.2 Wald Confidence Intervals for Model Parameters 147

8.3.3 Wald Confidence Intervals for Functions of μ and σ 148

8.4 The ML Estimate May Not Go Through the Points 151

8.5 Estimation with a Given Shape Parameter 152

8.5.1 Estimation for a Weibull/Smallest Extreme Value Distribution With Given σ 152

8.5.2 Estimation for a Weibull/Smallest Extreme Value Distribution With Given β = 1/σ and Zero Failures 155

9 Parametric Bootstrap and Other Simulation-Based Confidence Interval Methods 164

9.1 Introduction 165

9.1.1 Motivation 165

9.1.2 Basic Concepts 165

9.2 Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 165

9.2.1 Bootstrap Resampling 166

9.2.2 Fractional-Random-Weight Bootstrap Sampling 166

9.2.3 Parametric Bootstrap Samples and Bootstrap Estimates 169

9.2.4 How to Choose Which Bootstrap Sampling Method to Use 169

9.2.5 Choosing the Number of Bootstrap Samples 170

9.3 Bootstrap Confidence Interval Methods 171

9.3.1 Calculation of Quantiles of a Bootstrap Distribution 171

9.3.2 The Simple Percentile Method 171

9.3.3 The BC Percentile Method 173

9.3.4 The Bootstrap-t Method 174

9.4 Bootstrap Confidence Intervals Based on Pivotal Quantities 176

9.4.1 Introduction 176

9.4.2 Pivotal Quantity Confidence Intervals for the Location Parameter of a Location-Scale Distribution or the Scale Parameter of a Log-Location-Scale Distribution 177

9.4.3 Pivotal Quantity Confidence Intervals for the Scale Parameter of a Location-Scale Distribution or the Shape Parameter of a Log-Location-Scale Distribution 178

9.4.4 Pivotal Quantity Confidence Intervals for the p Quantile of a Location-Scale or a Log Location-Scale Distribution 179

9.5 Confidence Intervals Based on Generalized Pivotal Quantities 181

9.5.1 Generalized Pivotal Quantities for μ and σ of a Location-Scale Distribution and for Functions of μ and σ 181

9.5.2 Confidence Intervals for Tail Probabilities for (Log-)Location-Scale Distributions 182

9.5.3 Confidence Intervals for the Mean of a Log-Location-Scale Distribution 183

10 An Introduction to Bayesian Statistical Methods for Reliability 189

10.1 Bayesian Inference: Overview 190

10.1.1 Motivation 190

10.1.2 The Relationship between Non-Bayesian Likelihood Inference and Bayesian Inference 190

10.1.3 Bayes’ Theorem and Bayesian Data Analysis 191

10.1.4 The Need for Prior Information 192

10.1.5 Parameterization 192

10.2 Bayesian Inference: an Illustrative Example 194

10.2.1 Specification of Prior Information 194

10.2.2 Characterizing the Joint Posterior Distribution via Simulation 195

10.2.3 Comparison of Joint Posterior Distributions Based on Weakly Informative and Informative Prior Information on the Weibull Shape Parameter β 195

10.2.4 Generating Sample Draws Via Simple Simulation 196

10.2.5 Using the Sample Draws to Construct Bayesian Point Estimates and Credible Intervals 197

10.3 More About Prior Information and Specification of a Prior Distribution 202

10.3.1 Noninformative Prior Distributions 202

10.3.2 Weakly Informative and Informative Prior Distributions 203

10.3.3 Using a Range to Specify a Prior Distribution 203

10.3.4 Whose Prior Distribution Should We Use? 204

10.3.5 Sources of Prior Information 205

10.4 Implementing Bayesian Analyses Using MCMC Simulation 205

10.4.1 Basic Ideas of MCMC Simulation 205

10.4.2 Risks of Misuse and Diagnostics 206

10.4.3 MCMC Summary 208

10.4.4 Software for MCMC 210

10.5 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 210

10.5.1 Background 210

10.5.2 Rocket-Motor Prior Information 212

10.5.3 Rocket-Motor Bayesian Estimation Results 212

10.5.4 Credible Interval for the Proportion of Healthy Rocket Motors after 20 or 30 Years in the Stockpile 213

11 Special Parametric Models 219

11.1 Extending ML Methods 219

11.1.1 Likelihood for Other Distributions and Models 219

11.1.2 Confidence Intervals for Other Distributions and Models 220

11.2 Fitting the Generalized Gamma Distribution 220

11.3 Fitting the Birnbaum–Saunders Distribution 223

11.3.1 Birnbaum–Saunders Distribution 223

11.3.2 Birnbaum–Saunders ML Estimation 223

11.4 The Limited Failure Population Model 225

11.4.1 The LFP Likelihood Function and Its Maximum 225

11.4.2 Profile Likelihood Functions and LR-Based Confidence Intervals for μ, σ, and p 226

11.5 Truncated Data and Truncated Distributions 227

11.5.1 Examples of Left Truncation 227

11.5.2 Likelihood with Left Truncation 228

11.5.3 Nonparametric Estimation with Left Truncation 229

11.5.4 ML Estimation with Left Truncated Data 229

11.5.5 Examples of Right Truncation 230

11.5.6 Likelihood with Right (and Left) Truncation 231

11.5.7 Nonparametric Estimation with Right (and Left) Truncation 231

11.5.8 A Trick to Handle Truncated Observations 231

11.6 Fitting Distributions that Have a Threshold Parameter 232

11.6.1 Estimation with a Given Threshold Parameter 232

11.6.2 Probability Plotting Methods 232

11.6.3 Likelihood Methods 233

11.6.4 Summary of Results of Fitting Models to Skewed Distributions 236

12 Comparing Failure-Time Distributions 243

12.1 Background and Motivation 243

12.1.1 Reasons for comparing failure-time distributions 243

12.1.2 Motivating examples 244

12.2 Nonparametric Comparisons 244

12.2.1 Graphical nonparametric comparisons 244

12.2.2 Nonparametric comparison tests 244

12.3 Parametric Comparison of Two Groups by Fitting Separate Distributions 247

12.4 Parametric Comparison of Two Groups by Fitting Separate Distributions With Equal σ values 248

12.5 Parametric Comparison of More than Two Groups 250

12.5.1 Comparison Using Separate Analyses 250

12.5.2 Comparison Using Equal-σ Values 251

12.5.3 Comparison Using Simultaneous Confidence Intervals 253

13 Planning Life Tests for Estimation 261

13.1 Introduction 261

13.1.1 Basic Ideas 261

13.2 Simple Formulas to Determine the Needed Sample Size 263

13.2.1 Motivation for Use of Large-Sample Approximations of Test Plan Properties 263

13.2.2 Estimating an Unrestricted Quantile and Other Unrestricted Quantities 263

13.2.3 Plots of Quantile Variance Factors 264

13.2.4 Sample Size Formula for Estimating an Unrestricted Quantile and Other Unrestricted Quantities 264

13.2.5 Estimating a Positive Quantile and Other Positive Quantities 266

13.2.6 Sample Size Formula for Estimating a Positive Quantile and Other Positive Quantities 266

13.2.7 Meeting the Precision Criterion 267

13.3 Use of Simulation in Test Planning 267

13.3.1 Basic Idea 267

13.3.2 Assessing the Effect of Test Length on Precision 267

13.3.3 Assessing the Tradeoff Between Sample Size and Test Length 272

13.3.4 Uncertainty in Planning Values 272

13.4 Approximate Variance of ML Estimators and Computing Variance Factors 274

13.4.1 A General Large-Sample Approximation for the Variances of ML Estimators 274

13.4.2 A General Large-Sample Approximation for the Variance of the ML Estimator of a Function of the Parameters 274

13.5 Variance Factors for (Log-)Location-Scale Distributions 275

13.5.1 Large-Sample Approximate Variance-Covariance Matrix for Location-Scale Parameters 275

13.5.2 Variance Factors for (Log-)Location-Scale Distribution Parameter Estimators 276

13.5.3 Variance Factors for Functions of (Log-)Location-Scale Distribution Parameter Estimators 277

13.5.4 Variance Factors to Estimate a Quantile When T is Log-Location-Scale (μ, σ) 277

13.6 Some Extensions 278

13.6.1 Type 2 (Failure) Censoring 278

13.6.2 Variance Factors for Location-Scale Parameters and Multiple Censoring 278

13.6.3 Test Planning for Distributions That Are Not Log-Location-Scale 279

14 Planning Reliability Demonstration Tests 282

14.1 Introduction to Demonstration Testing 282

14.1.1 Criteria for Doing a Demonstration 282

14.1.2 Basic Ideas of Demonstration Testing 283

14.1.3 Data and Distribution 283

14.1.4 The Important Relationship Between S(td) and S(tc) 283

14.1.5 The Demonstration Test Decision Rule 283

14.2 Finding the Required Sample Size n or Test-Length Factor k 284

14.2.1 Required Sample Size n for a Given Test-Length Factor k 284

14.2.2 Required Test-Length Factor k for a Given Sample Size n 284

14.2.3 Minimum-Sample-Size Test 284

14.2.4 Minimum-Sample-Size Test for the Weibull Distribution 284

14.3 Probability of Successful Demonstration 288

14.3.1 General Approach 288

14.3.2 Special Result for the Weibull Minimum Sample Size Test 288

15 Prediction of Failure Times and the Number of Future Field Failures 293

15.1 Basic Concepts of Statistical Prediction 294

15.1.1 Motivation and Prediction Applications 294

15.1.2 What is Needed to Compute a Prediction Interval? 295

15.2 Probability Prediction Intervals (_ Known) 295

15.3 Statistical Prediction Intervals (_ Estimated) 296

15.3.1 Coverage Probability Concepts 296

15.3.2 Relationship Between One-Sided Prediction Bounds and Two-Sided Prediction Intervals 296

15.3.3 Prediction Based on a Pivotal Quantity 297

15.4 Plug-In Prediction and Calibration 297

15.4.1 The Plug-In Method for Computing an Approximate Statistical Prediction Interval 297

15.4.2 Calibrating Plug-In Statistical Prediction Bounds 299

15.4.3 The Calibration-Bootstrap Prediction Method 299

15.4.4 Finding a Calibration Curve by Computing Coverage Probabilities for the Plug-In Method 300

15.4.5 Assessing the Amount of Monte Carlo Error 301

15.5 Computing and Using Predictive Distributions 301

15.5.1 Definition and Use of a Predictive Distribution 301

15.5.2 A Simple Method for Computing a Predictive Distribution 302

15.5.3 Alternative Methods for Computing a Predictive Distribution 302

15.5.4 A General AlternativeMethod of Computing Prediction Intervals Using Calibration-Bootstrap and an Extra Layer of Simulation 304

15.6 Prediction of the Number of Future Failures from a Single Group of Units in the Field 304

15.6.1 Problem Background 304

15.6.2 Distribution of the Predictand, Point Prediction, and the Plug-in Prediction Method305

15.6.3 Correcting the Plug-in Method 306

15.7 Predicting the Number of Future Failures from Multiple Groups of Units in the Field with Staggered Entry into the Field 307

15.7.1 Distribution of the Number of Future Failures 308

15.7.2 Plug-in Prediction Bounds and Intervals for the Number of Future Failures 308

15.7.3 Approximations for the Poisson–Binomial Distribution 310

15.7.4 Improved Prediction Bounds and Intervals for the Number of Future Failures 310

15.8 Bayesian Prediction Methods 311

15.8.1 Motivation for the use of Bayesian Prediction Methods 311

15.8.2 Computing a Bayesian Predictive Distribution 311

15.9 Choosing a Distribution for Making Predictions 313

16 Analysis of Data with More than One Failure Mode 321

16.1 An Introduction to Multiple Failure Modes 321

16.1.1 Basic Idea 321

16.1.2 Multiple Failure Modes Data 322

16.2 Model for Multiple Failure Modes Data 323

16.2.1 Association Between Failure Times of Different Failure Modes 323

16.2.2 The Assumption of Independence 324

16.2.3 System Failure-Time Distribution with All Failure Modes Active 324

16.3 Competing-Risk Estimation 324

16.3.1 Maximum Likelihood Estimation with Multiple Failure Modes 324

16.3.2 Importance of Accounting for Failure-Mode Information 328

16.4 The Effect of Eliminating a Failure Mode 328

16.5 Subdistribution Functions and Prediction for Individual Failure Modes 331

16.5.1 Subdistribution Functions 331

16.5.2 Predictions for Individual Failure Modes 332

16.6 More About the Non-Identifiability of Dependence Among Failure Modes 332

17 Failure-Time Regression Analysis 340

17.1 Introduction 341

17.1.1 Motivating Example 341

17.1.2 Failure-time Regression Models 341

17.2 Simple Linear Regression Models 342

17.2.1 Location-Scale Regression Model and Likelihood 342

17.2.2 Log-Location-Scale Regression Model and Likelihood 343

17.3 Standard Errors and Confidence Intervals for Regression Models 345

17.3.1 Standard Errors and Confidence Intervals for Parameters 345

17.3.2 Standard Errors and Confidence Intervals for Distribution Quantities at Specific Explanatory Variable Conditions 346

17.4 Regression Model with Quadratic μ and Nonconstant σ 347

17.4.1 Quadratic Regression Relationship for μ and a Constant σ Parameter 348

17.4.2 Quadratic Regression Model with Nonconstant Shape Parameter σ 349

17.4.3 Further Comments on the Use of Empirical Regression Models 350

17.4.4 Comments on Numerical Methods and Parameterization 350

17.5 Checking Model Assumptions 351

17.5.1 Definition of Residuals 351

17.5.2 Cox–Snell Residuals 351

17.5.3 Regression Diagnostics 352

17.6 Empirical Regression Models and Sensitivity Analysis 354

17.7 Models with Two or More Explanatory Variables 359

17.7.1 Model-Free Graphical Analysis of Two-Variable Regression Data 359

17.7.2 Two-Variable Regression Model without Interaction 359

17.7.3 Two-Variable Regression Model with Interaction 361

18 Analysis of Accelerated Life Test Data 369

18.1 Introduction to Accelerated Life Tests 369

18.1.1 Motivation and Background for Accelerated Testing 369

18.1.2 Different Methods of Acceleration 370

18.2 Overview of ALT Data Analysis Methods 371

18.2.1 ALT Models 371

18.2.2 Strategy for Analyzing ALT Data 371

18.3 Temperature-Accelerated Life Tests 372

18.3.1 Introduction 372

18.3.2 Scatterplot of ALT Data 372

18.3.3 The Arrhenius Acceleration Model 375

18.3.4 Checking Other Model Assumptions 377

18.3.5 ML Estimates at Use Conditions 378

18.4 Bayesian Analysis of a Temperature-Accelerated Life Test 380

18.4.1 Introduction 380

18.4.2 Parameterization of the Arrhenius Model 380

18.4.3 Prior Distribution Specification in an ALT 380

18.4.4 Bayesian Analysis of the Device-A ALT Data 381

18.5 Voltage-Accelerated Life Test 381

18.5.1 Voltage and Voltage-Stress Acceleration 382

18.5.2 The Inverse-Power Relationship 383

18.5.3 ML Estimates at Use Conditions for the M-P Insulation 388

18.5.4 Physical Motivation for the Inverse-Power Relationship for Voltage-Stress Acceleration 388

18.5.5 A Generalization of the Inverse Power Relationship 389

19 More Topics on Accelerated Life Testing 396

19.1 ALTs with Interval-Censored Data 396

19.1.1 ML Estimation at Individual Test Conditions 397

19.1.2 ML Estimates of the Arrhenius-LognormalModel Parameters with Interval-Censored Data 398

19.1.3 Fitting an ALT Model with a Given Relationship Slope 399

19.1.4 Bayesian Analysis of Interval Censored ALT Data 400

19.2 ALTs with Two Accelerating Variables 401

19.3 Multifactor Experiments with a Single Accelerating Variable 405

19.4 Practical Suggestions for Drawing Conclusions from ALT Data 409

19.4.1 Predicting Product Performance 409

19.4.2 Drawing Conclusions from ALTs 409

19.4.3 Planning ALTs 410

19.5 Pitfalls of Accelerated Life Testing 410

19.5.1 Pitfall: Extraneous Failure Modes Caused by Too Much Acceleration 410

19.5.2 Pitfall: Masked Failure Modes 411

19.5.3 Pitfall: Faulty Comparison 411

19.6 Other Kinds of Accelerated Tests 412

19.6.1 Accelerated Tests with Step- and Varying-Stress 412

19.6.2 Continuous Product Operation Product to Accelerate Testing 413

19.6.3 Qualitative Accelerated Life Tests 414

19.6.4 Burn-in 414

20 Degradation Modeling and Destructive Degradation Data Analysis 421

20.1 Degradation Reliability Data and Degradation Path Models: Introduction and Background422

20.1.1 Motivation 422

20.1.2 Examples of Degradation Data 422

20.1.3 Limitations of Degradation Data 423

20.2 Description and Mechanistic Motivation for Degradation Path Models 423

20.2.1 Shapes of Degradation Paths 423

20.2.2 A Statistical Model for Degradation Data without Explanatory Variables 425

20.2.3 A Statistical Model for Degradation Data with Explanatory Variables 425

20.2.4 Degradation Path Models 425

20.3 Models Relating Degradation and Failure 427

20.3.1 Soft Failures: Specified Degradation Level 427

20.3.2 Hard Failures: Joint Distribution of Degradation and Failure Level 427

20.4 DDT Background, Motivating Examples, and Estimation 427

20.4.1 Background 427

20.4.2 Motivating examples 427

20.4.3 Transformations for ADDT Data 429

20.4.4 Fitting a Statistical Model to ADDT Data 429

20.4.5 Degradation Model Checking 431

20.5 Failure-Time Distributions Induced from DDT Models and Failure-Time Inferences 431

20.5.1 A General Approach to Obtaining the Failure-Time Distribution for DDT Models 431

20.5.2 Failure-Time Inferences for Model 2 432

20.6 ADDT Model Building 433

20.6.1 Transformations for ADDT Data 433

20.6.2 Fitting Separate Models to the Different Levels of the Accelerating Variable 434

20.7 Fitting an Acceleration Model to ADDT Data 435

20.7.1 A Model and Likelihood for ADDT Data 435

20.7.2 ADDT Model Checking 435

20.8 ADDT Failure-Time Inferences 437

20.8.1 Failure-Time cdf for Model 6 437

20.8.2 Failure-Time Distribution Quantiles for Model 6 438

20.9 ADDT Analysis Using an Informative Prior Distribution 438

20.10 An ADDT with an Asymptotic Model 439

20.10.1 ADDT Data With an Asymptote 441

20.10.2 Finding a Model for ADDT Data with an Asymptote 441

20.10.3 Fitting an ADDT Model with an Asymptote 442

20.10.4 ADDT Model Checking with an Asymptotic Model 443

20.10.5 Failure-Time cdf for Model 8 444

20.10.6 Failure-Time Distribution Quantiles for Model 8 444

21 Repeated-Measures Degradation Modeling and Analysis 448

21.1 RMDT Models and Data 448

21.1.1 RMDT Motivating Example 448

21.1.2 Repeated-Measures Degradation Models 449

21.1.3 Models for Variation in Degradation and Failure Times 450

21.2 RMDT Parameter Estimation 451

21.2.1 RMDT Models with Random Parameters 451

21.2.2 The Likelihood for Random-Parameter Models 452

21.2.3 Bayesian Estimation with Random Parameters 452

21.2.4 RMDT Modeling and Diagnostics 453

21.3 The Relationship Between Degradation and Failure-Time for RMDT Models 454

21.3.1 Time-to-First-Crossing Distribution 454

21.3.2 A General Approach 454

21.3.3 Analytical Solution for F(t) 454

21.3.4 Numerical Evaluation of F(t) 456

21.3.5 Monte Carlo Evaluation of F(t) 456

21.4 Estimation of a Failure-Time cdf from RMDT Data 457

21.5 Models for ARMDT Data 458

21.6 ARMDT Estimation 459

21.6.1 Estimation of Failure Probabilities, Distribution Quantiles, and Other Functions of Model Parameters for an ARMDT Model 459

21.6.2 ARMDT Analysis Using an Informative Prior Distribution 460

21.7 ARMDT with Multiple Accelerating Variables 462

22 Analysis of Repairable System and Other Recurrent Events Data 469

22.1 Introduction 469

22.1.1 Recurrent Events Data 469

22.1.2 A Nonparametric Model for Recurrent Events Data 470

22.2 Nonparametric Estimation of the MCF 471

22.2.1 Nonparametric Model Assumptions 471

22.2.2 Point Estimate of the MCF 471

22.2.3 Confidence Intervals for _ 472

22.3 Comparison of Two Samples of Recurrent Events Data 474

22.4 Recurrent Events Data with Multiple Event Types 475

23 Case Studies and Further Applications 481

23.1 Analysis of Hard Drive Field Data 481

23.1.1 Data and background 481

23.1.2 The GLFP Model 482

23.1.3 GLFP Likelihood for the Backblaze-14 Data 482

23.1.4 Bayesian Estimation of the Backblaze-14 GLFP Model Parameters 483

23.2 Reliability in the Presence of Stress-Strength Interference 484

23.2.1 Definition of Stress-Strength Reliability 484

23.2.2 Distributions of Stress and Strength 484

23.2.3 ML Estimates and Confidence Intervals for Stress and Strength Reliability 486

23.2.4 Bayesian Estimation for Stress and Strength Reliability 486

23.3 Predicting Field Failures with a Limited Failure Population 487

23.3.1 ML Analysis of the Device-J Field Data 488

23.3.2 Bayesian Prediction for the Number of Future Device-J Failures 490

23.4 Analysis of Accelerated Life Test Data When There is a Batch Effect 494

23.4.1 Kevlar Pressure Vessels Background and Data 494

23.4.2 Model for the Kevlar Pressure Vessels ALT Data 494

23.4.3 Bayesian Estimation and Reliability Inferences 495

23.4.4 Bayesian Estimation of System Reliability 497

Epilogue 499

A Notation and Acronyms 503

B Other Useful Distributions and Probability Distribution Computations 509

B.1 Important Characteristics of Distribution Functions 509

B.1.1 Density and Probability Mass Functions 510

B.1.2 Cumulative Distribution Function 510

B.1.3 Quantile Function 510

B.2 Distributions and R Computations 511

B.3 Continuous Distributions 511

B.3.1 Common Location-Scale and Log-Location-Scale Distributions 511

B.3.2 Beta Distribution 514

B.3.3 Uniform Distribution 514

B.3.4 Loguniform Distribution 514

B.3.5 Gamma Distribution 515

B.3.6 Chi-Square Distribution 515

B.3.7 Truncated Normal Distribution 515

B.3.8 Student’s t-Distribution 516

B.3.9 Location-Scale t-Distribution 516

B.3.10 Half Location-Scale t-Distribution 517

B.3.11 Bivariate Normal Distribution 517

B.3.12 Dirichlet Distribution 518

B.4 Discrete Distributions 519

B.4.1 Binomial Distribution 519

B.4.2 Poisson Distribution 520

B.4.3 Poisson–binomial Distribution 520

C Some Results from Statistical Theory 522

C.1 The cdfs and pdfs of Functions of Random Variables 522

C.1.1 Transformation of Continuous Random Variables 523

C.2 Statistical Error Propagation—The Delta Method 527

C.3 Likelihood and Fisher Information Matrices 528

C.4 Regularity Conditions 529

C.4.1 Regularity Conditions for Location-Scale Distributions 529

C.4.2 General Regularity Conditions 529

C.4.3 Asymptotic Theory for Nonregular Models 530

C.5 Convergence in Distribution 530

C.5.1 Central Limit Theorem and Other Examples of Convergence in Distribution 530

C.6 Convergence in Probability 531

C.7 Outline of General ML Theory 532

C.7.1 Asymptotic Distribution of ML Estimators 532

C.7.2 Asymptotic Covariance Matrix for Test Planning 532

C.7.3 Asymptotic Distribution of Functions of ML Estimators 532

C.7.4 Estimating the Variance–Covariance Matrix of ML Estimates 533

C.7.5 Likelihood Ratios and Profile Likelihoods 533

C.7.6 Approximate Likelihood-Ratio-Based Confidence Regions or Confidence Intervals for the Model Parameters 533

C.7.7 Approximate Confidence Regions and Intervals Based on Asymptotic Normality of ML Estimators 534

C.8 Inference with Zero or Few Failures 534

C.8.1 Exponential Distribution Inference with Zero or Few Failures 534

C.8.2 Weibull Distribution Inference with Given β and Zero or Few Failures 536

C.9 The Bonferroni Inequality 536

D Tables 538

References 549