Statistical Methods for Reliability Data
, by Meeker, William Q.; Escobar, Luis A.; Pascual, Francis G.
- ISBN: 9781118115459 | 1118115457
- Cover: Hardcover
- Copyright: 12/29/2021
An authoritative guide to the most recent advances in statistical methods for quantifying reliability
Statistical Methods for Reliability Data, Second Edition (SMRD2) is an essential guide to the most widely used and recently developed statistical methods for reliability data analysis and reliability test planning. Written by three experts in the area, SMRD2 updates and extends the long- established statistical techniques and shows how to apply powerful graphical, numerical, and simulation-based methods to a range of applications in reliability. SMRD2 is a comprehensive resource that describes maximum likelihood and Bayesian methods for solving practical problems that arise in product reliability and similar areas of application. SMRD2 illustrates methods with numerous applications and all the data sets are available on the book’s website. Also, SMRD2 contains an extensive collection of exercises that will enhance its use as a course textbook.
The SMRD2's website contains valuable resources, including R packages, Stan model codes, presentation slides, technical notes, information about commercial software for reliability data analysis, and csv files for the 93 data sets used in the book's examples and exercises. The importance of statistical methods in the area of engineering reliability continues to grow and SMRD2 offers an updated guide for, exploring, modeling, and drawing conclusions from reliability data.
SMRD2 features:
- Contains a wealth of information on modern methods and techniques for reliability data analysis
- Offers discussions on the practical problem-solving power of various Bayesian inference methods
- Provides examples of Bayesian data analysis performed using the R interface to the Stan system based on Stan models that are available on the book's website
- Includes helpful technical-problem and data-analysis exercise sets at the end of every chapter
- Presents illustrative computer graphics that highlight data, results of analyses, and technical concepts
Written for engineers and statisticians in industry and academia, Statistical Methods for Reliability Data, Second Edition offers an authoritative guide to this important topic.
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
What is included with this book?
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.
Please wait while the item is added to your bag...
×
Digital License
You are licensing a digital product for a set duration. Durations are set forth in the product description, with "Lifetime" typically meaning five (5) years of online access and permanent download to a supported device. All licenses are non-transferable.
More details can be found here.