Statistical Methods for Reliability Data (Wiley Series in Probability and Statistics)

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 theta 123 7.4 Wald (Normal-Approximation) Confidence Intervals for theta 125 7.5 Confidence Intervals for Functions of theta 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 sigma 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 mu and Nonconstant sigma 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