Statistical Methods for Reliability Data

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 iPreface to the Second Edition iiiPreface to First Edition viiiAcknowledgments xii1 Reliability Concepts and an Introduction to Reliability Data 11.1 Introduction 11.2 Examples of Reliability Data 31.3 General Models for Reliability Data 111.4 Models for Time to Event Versus Models for Recurrences in a Sequence of Events 131.5 Strategy for Data Collection, Modeling, and Analysis 152 Models, Censoring, and Likelihood for Failure-Time Data 192.1 Models for Continuous Failure-Time Processes 192.2 Models for Discrete Data from a Continuous Process 252.3 Censoring 272.4 Likelihood 283 Nonparametric Estimation for Failure-Time Data 373.1 Estimation from Complete Data 383.2 Estimation from Singly-Censored Interval Data 383.3 Basic Ideas of Statistical Inference 403.4 Confidence Intervals from Complete or Singly-Censored Data 413.5 Estimation from Multiply-Censored Data 433.6 Pointwise Confidence Intervals from Multiply-Censored Data 453.7 Estimation from Multiply-Censored Data with Exact Failures 473.8 Nonparametric Simultaneous Confidence Bands 493.9 Arbitrary Censoring 524 Some Parametric Distributions Used in Reliability Applications 604.1 Introduction 614.2 Quantities of Interest in Reliability Applications 614.3 Location-Scale and Log-Location-Scale Distributions 624.4 Exponential Distribution 634.5 Normal Distribution 644.6 Lognormal Distribution 654.7 Smallest Extreme Value Distribution 674.8 Weibull Distribution 684.9 Largest Extreme Value Distribution 704.10 Frechet Distribution 714.11 Logistic Distribution 734.12 Loglogistic Distribution 744.13 Generalized Gamma Distribution 754.14 Distributions with a Threshold Parameter 764.15 Other Methods of Deriving Failure-Time Distributions 784.16 Parameters and Parameterization 804.17 Generating Pseudorandom Observations from a Specified Distribution 805 System Reliability Concepts and Methods 875.1 Non-Repairable System Reliability Metrics 885.2 Series Systems 885.3 Parallel Systems 915.4 Series-Parallel Systems 935.5 Other System Structures 945.6 Multistate System Reliability Models 966 Probability Plotting 1026.1 Introduction 1036.2 Linearizing Location-Scale-Based Distributions 1036.3 Graphical Goodness of Fit 1056.4 Probability Plotting Positions 1066.5 Notes on the Application of Probability Plotting 1117 Parametric Likelihood Fitting Concepts: Exponential Distribution 1197.1 Introduction 1207.2 Parametric Likelihood 1227.3 Likelihood Confidence Intervals for θ 1237.4 Wald (Normal-Approximation) Confidence Intervals for θ 1257.5 Confidence Intervals for Functions of θ 1267.6 Comparison of Confidence Interval Procedures 1277.7 Likelihood for Exact Failure Times 1287.8 Effect of Sample Size on Confidence Interval Width and the Likelihood Shape 1307.9 Exponential Distribution Inferences with No Failures 1318 Maximum Likelihood Estimation for Log-Location-Scale Distributions 1388.1 Likelihood Definition 1398.2 Likelihood Confidence Regions and Intervals 1428.3 Wald Confidence Intervals 1468.4 The ML Estimate May Not Go Through the Points 1518.5 Estimation with a Given Shape Parameter 1529 Parametric Bootstrap and Other Simulation-Based Confidence Interval Methods 1649.1 Introduction 1659.2 Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 1659.3 Bootstrap Confidence Interval Methods 1719.4 Bootstrap Confidence Intervals Based on Pivotal Quantities 1769.5 Confidence Intervals Based on Generalized Pivotal Quantities 18110 An Introduction to Bayesian Statistical Methods for Reliability 18910.1 Bayesian Inference: Overview 19010.2 Bayesian Inference: an Illustrative Example 19410.3 More About Prior Information and Specification of a Prior Distribution 20210.4 Implementing Bayesian Analyses Using MCMC Simulation 20510.5 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 21011 Special Parametric Models 21911.1 Extending ML Methods 21911.2 Fitting the Generalized Gamma Distribution 22011.3 Fitting the Birnbaum–Saunders Distribution 22311.4 The Limited Failure Population Model 22511.5 Truncated Data and Truncated Distributions 22711.6 Fitting Distributions that Have a Threshold Parameter 23212 Comparing Failure-Time Distributions 24312.1 Background and Motivation 24312.2 Nonparametric Comparisons 24412.3 Parametric Comparison of Two Groups by Fitting Separate Distributions 24712.4 Parametric Comparison of Two Groups by Fitting Separate Distributions With Equal σ values 24812.5 Parametric Comparison of More than Two Groups 25013 Planning Life Tests for Estimation 26113.1 Introduction 26113.2 Simple Formulas to Determine the Needed Sample Size 26313.3 Use of Simulation in Test Planning 26713.4 Approximate Variance of ML Estimators and Computing Variance Factors 27413.5 Variance Factors for (Log-)Location-Scale Distributions 27513.6 Some Extensions 27814 Planning Reliability Demonstration Tests 28214.1 Introduction to Demonstration Testing 28214.2 Finding the Required Sample Size n or Test-Length Factor k 28414.3 Probability of Successful Demonstration 28815 Prediction of Failure Times and the Number of Future Field Failures 29315.1 Basic Concepts of Statistical Prediction 29415.2 Probability Prediction Intervals (_ Known) 29515.3 Statistical Prediction Intervals (_ Estimated) 29615.4 Plug-In Prediction and Calibration 29715.5 Computing and Using Predictive Distributions 30115.6 Prediction of the Number of Future Failures from a Single Group of Units in the Field 30415.7 Predicting the Number of Future Failures from Multiple Groups of Units in the Field with Staggered Entry into the Field 30715.8 Bayesian Prediction Methods 31115.9 Choosing a Distribution for Making Predictions 31316 Analysis of Data with More than One Failure Mode 32116.1 An Introduction to Multiple Failure Modes 32116.2 Model for Multiple Failure Modes Data 32316.3 Competing-Risk Estimation 32416.4 The Effect of Eliminating a Failure Mode 32816.5 Subdistribution Functions and Prediction for Individual Failure Modes 33116.6 More About the Non-Identifiability of Dependence Among Failure Modes 33217 Failure-Time Regression Analysis 34017.1 Introduction 34117.2 Simple Linear Regression Models 34217.3 Standard Errors and Confidence Intervals for Regression Models 34517.4 Regression Model with Quadratic μ and Nonconstant σ 34717.5 Checking Model Assumptions 35117.6 Empirical Regression Models and Sensitivity Analysis 35417.7 Models with Two or More Explanatory Variables 35918 Analysis of Accelerated Life Test Data 36918.1 Introduction to Accelerated Life Tests 36918.2 Overview of ALT Data Analysis Methods 37118.3 Temperature-Accelerated Life Tests 37218.4 Bayesian Analysis of a Temperature-Accelerated Life Test 38018.5 Voltage-Accelerated Life Test 38119 More Topics on Accelerated Life Testing 39619.1 ALTs with Interval-Censored Data 39619.2 ALTs with Two Accelerating Variables 40119.3 Multifactor Experiments with a Single Accelerating Variable 40519.4 Practical Suggestions for Drawing Conclusions from ALT Data 40919.5 Pitfalls of Accelerated Life Testing 41019.6 Other Kinds of Accelerated Tests 41220 Degradation Modeling and Destructive Degradation Data Analysis 42120.1 Degradation Reliability Data and Degradation Path Models: Introduction and Background42220.2 Description and Mechanistic Motivation for Degradation Path Models 42320.3 Models Relating Degradation and Failure 42720.4 DDT Background, Motivating Examples, and Estimation 42720.5 Failure-Time Distributions Induced from DDT Models and Failure-Time Inferences 43120.6 ADDT Model Building 43320.7 Fitting an Acceleration Model to ADDT Data 43520.8 ADDT Failure-Time Inferences 43720.9 ADDT Analysis Using an Informative Prior Distribution 43820.10 An ADDT with an Asymptotic Model 43921 Repeated-Measures Degradation Modeling and Analysis 44821.1 RMDT Models and Data 44821.2 RMDT Parameter Estimation 45121.3 The Relationship Between Degradation and Failure-Time for RMDT Models 45421.4 Estimation of a Failure-Time cdf from RMDT Data 45721.5 Models for ARMDT Data 45821.6 ARMDT Estimation 45921.7 ARMDT with Multiple Accelerating Variables 46222 Analysis of Repairable System and Other Recurrent Events Data 46922.1 Introduction 46922.2 Nonparametric Estimation of the MCF 47122.3 Comparison of Two Samples of Recurrent Events Data 47422.4 Recurrent Events Data with Multiple Event Types 47523 Case Studies and Further Applications 48123.1 Analysis of Hard Drive Field Data 48123.2 Reliability in the Presence of Stress-Strength Interference 48423.3 Predicting Field Failures with a Limited Failure Population 48723.4 Analysis of Accelerated Life Test Data When There is a Batch Effect 494Epilogue 499A Notation and Acronyms 503B Other Useful Distributions and Probability Distribution Computations 509B.1 Important Characteristics of Distribution Functions 509B.2 Distributions and R Computations 511B.3 Continuous Distributions 511B.4 Discrete Distributions 519B.4.1 Binomial Distribution 519C Some Results from Statistical Theory 522C.1 The cdfs and pdfs of Functions of Random Variables 522C.2 Statistical Error Propagation—The Delta Method 527C.3 Likelihood and Fisher Information Matrices 528C.4 Regularity Conditions 529C.5 Convergence in Distribution 530C.6 Convergence in Probability 531C.7 Outline of General ML Theory 532C.8 Inference with Zero or Few Failures 534C.9 The Bonferroni Inequality 536D Tables 538References 549

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