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
1.2 Examples of Reliability Data 3
1.3 General Models for Reliability Data 11
1.4 Models for Time to Event Versus Models for Recurrences in a Sequence of Events 13
1.5 Strategy for Data Collection, Modeling, and Analysis 15
2 Models, Censoring, and Likelihood for Failure-Time Data 19
2.1 Models for Continuous Failure-Time Processes 19
2.2 Models for Discrete Data from a Continuous Process 25
2.3 Censoring 27
2.4 Likelihood 28
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
3.4 Confidence Intervals from Complete or Singly-Censored Data 41
3.5 Estimation from Multiply-Censored Data 43
3.6 Pointwise Confidence Intervals from Multiply-Censored Data 45
3.7 Estimation from Multiply-Censored Data with Exact Failures 47
3.8 Nonparametric Simultaneous Confidence Bands 49
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
4.5 Normal Distribution 64
4.6 Lognormal Distribution 65
4.7 Smallest Extreme Value Distribution 67
4.8 Weibull Distribution 68
4.9 Largest Extreme Value Distribution 70
4.10 Frechet Distribution 71
4.11 Logistic Distribution 73
4.12 Loglogistic Distribution 74
4.13 Generalized Gamma Distribution 75
4.14 Distributions with a Threshold Parameter 76
4.15 Other Methods of Deriving Failure-Time Distributions 78
4.16 Parameters and Parameterization 80
4.17 Generating Pseudorandom Observations from a Specified Distribution 80
5 System Reliability Concepts and Methods 87
5.1 Non-Repairable System Reliability Metrics 88
5.2 Series Systems 88
5.3 Parallel Systems 91
5.4 Series-Parallel Systems 93
5.5 Other System Structures 94
5.6 Multistate System Reliability Models 96
6 Probability Plotting 102
6.1 Introduction 103
6.2 Linearizing Location-Scale-Based Distributions 103
6.3 Graphical Goodness of Fit 105
6.4 Probability Plotting Positions 106
6.5 Notes on the Application of Probability Plotting 111
7 Parametric Likelihood Fitting Concepts: Exponential Distribution 119
7.1 Introduction 120
7.2 Parametric Likelihood 122
7.3 Likelihood Confidence Intervals for θ 123
7.4 Wald (Normal-Approximation) Confidence Intervals for θ 125
7.5 Confidence Intervals for Functions of θ 126
7.6 Comparison of Confidence Interval Procedures 127
7.7 Likelihood for Exact Failure Times 128
7.8 Effect of Sample Size on Confidence Interval Width and 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
8.2 Likelihood Confidence Regions and Intervals 142
8.3 Wald Confidence Intervals 146
8.4 The ML Estimate May Not Go Through the Points 151
8.5 Estimation with a Given Shape Parameter 152
9 Parametric Bootstrap and Other Simulation-Based Confidence Interval Methods 164
9.1 Introduction 165
9.2 Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 165
9.3 Bootstrap Confidence Interval Methods 171
9.4 Bootstrap Confidence Intervals Based on Pivotal Quantities 176
9.5 Confidence Intervals Based on Generalized Pivotal Quantities 181
10 An Introduction to Bayesian Statistical Methods for Reliability 189
10.1 Bayesian Inference: Overview 190
10.2 Bayesian Inference: an Illustrative Example 194
10.3 More About Prior Information and Specification of a Prior Distribution 202
10.4 Implementing Bayesian Analyses Using MCMC Simulation 205
10.5 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 210
11 Special Parametric Models 219
11.1 Extending ML Methods 219
11.2 Fitting the Generalized Gamma Distribution 220
11.3 Fitting the Birnbaum–Saunders Distribution 223
11.4 The Limited Failure Population Model 225
11.5 Truncated Data and Truncated Distributions 227
11.6 Fitting Distributions that Have a Threshold Parameter 232
12 Comparing Failure-Time Distributions 243
12.1 Background and Motivation 243
12.2 Nonparametric Comparisons 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
13 Planning Life Tests for Estimation 261
13.1 Introduction 261
13.2 Simple Formulas to Determine the Needed Sample Size 263
13.3 Use of Simulation in Test Planning 267
13.4 Approximate Variance of ML Estimators and Computing Variance Factors 274
13.5 Variance Factors for (Log-)Location-Scale Distributions 275
13.6 Some Extensions 278
14 Planning Reliability Demonstration Tests 282
14.1 Introduction to Demonstration Testing 282
14.2 Finding the Required Sample Size n or Test-Length Factor k 284
14.3 Probability of Successful Demonstration 288
15 Prediction of Failure Times and the Number of Future Field Failures 293
15.1 Basic Concepts of Statistical Prediction 294
15.2 Probability Prediction Intervals (_ Known) 295
15.3 Statistical Prediction Intervals (_ Estimated) 296
15.4 Plug-In Prediction and Calibration 297
15.5 Computing and Using Predictive Distributions 301
15.6 Prediction of the Number of Future Failures from a Single Group of Units in the Field 304
15.7 Predicting the Number of Future Failures from Multiple Groups of Units in the Field with Staggered Entry into the Field 307
15.8 Bayesian Prediction Methods 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
16.2 Model for Multiple Failure Modes Data 323
16.3 Competing-Risk Estimation 324
16.4 The Effect of Eliminating a Failure Mode 328
16.5 Subdistribution Functions and Prediction for Individual Failure Modes 331
16.6 More About the Non-Identifiability of Dependence Among Failure Modes 332
17 Failure-Time Regression Analysis 340
17.1 Introduction 341
17.2 Simple Linear Regression Models 342
17.3 Standard Errors and Confidence Intervals for Regression Models 345
17.4 Regression Model with Quadratic μ and Nonconstant σ 347
17.5 Checking Model Assumptions 351
17.6 Empirical Regression Models and Sensitivity Analysis 354
17.7 Models with Two or More Explanatory Variables 359
18 Analysis of Accelerated Life Test Data 369
18.1 Introduction to Accelerated Life Tests 369
18.2 Overview of ALT Data Analysis Methods 371
18.3 Temperature-Accelerated Life Tests 372
18.4 Bayesian Analysis of a Temperature-Accelerated Life Test 380
18.5 Voltage-Accelerated Life Test 381
19 More Topics on Accelerated Life Testing 396
19.1 ALTs with Interval-Censored Data 396
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
19.5 Pitfalls of Accelerated Life Testing 410
19.6 Other Kinds of Accelerated Tests 412
20 Degradation Modeling and Destructive Degradation Data Analysis 421
20.1 Degradation Reliability Data and Degradation Path Models: Introduction and Background422
20.2 Description and Mechanistic Motivation for Degradation Path Models 423
20.3 Models Relating Degradation and Failure 427
20.4 DDT Background, Motivating Examples, and Estimation 427
20.5 Failure-Time Distributions Induced from DDT Models and Failure-Time Inferences 431
20.6 ADDT Model Building 433
20.7 Fitting an Acceleration Model to ADDT Data 435
20.8 ADDT Failure-Time Inferences 437
20.9 ADDT Analysis Using an Informative Prior Distribution 438
20.10 An ADDT with an Asymptotic Model 439
21 Repeated-Measures Degradation Modeling and Analysis 448
21.1 RMDT Models and Data 448
21.2 RMDT Parameter Estimation 451
21.3 The Relationship Between Degradation and Failure-Time for RMDT Models 454
21.4 Estimation of a Failure-Time cdf from RMDT Data 457
21.5 Models for ARMDT Data 458
21.6 ARMDT Estimation 459
21.7 ARMDT with Multiple Accelerating Variables 462
22 Analysis of Repairable System and Other Recurrent Events Data 469
22.1 Introduction 469
22.2 Nonparametric Estimation of the MCF 471
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
23.2 Reliability in the Presence of Stress-Strength Interference 484
23.3 Predicting Field Failures with a Limited Failure Population 487
23.4 Analysis of Accelerated Life Test Data When There is a Batch Effect 494
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.2 Distributions and R Computations 511
B.3 Continuous Distributions 511
B.4 Discrete Distributions 519
B.4.1 Binomial Distribution 519
C Some Results from Statistical Theory 522
C.1 The cdfs and pdfs of Functions of Random Variables 522
C.2 Statistical Error Propagation—The Delta Method 527
C.3 Likelihood and Fisher Information Matrices 528
C.4 Regularity Conditions 529
C.5 Convergence in Distribution 530
C.6 Convergence in Probability 531
C.7 Outline of General ML Theory 532
C.8 Inference with Zero or Few Failures 534
C.9 The Bonferroni Inequality 536
D Tables 538
References 549
