Reliability, Life Testing and the Prediction of Service Lives: For Engineers and Scientists (Springer Series in Statistics)

this Book Is Intended For Students And Practitioners Who Have Had A Calculus-based Statistics Course And Who Have An Interest In Safety Considerations Such As Reliability, Strength, And Duration-of-load Or Service Life. Many Persons Studying Statistical Science Will Be Employed Professionally Where The Problems Encountered Are Obscure, What Should Be Analyzed Is Not Clear, The Appropriate Assumptions Are Equivocal, And Data Are Scant. Yet Tutorial Problems Of This Nature Are Virtually Never Encountered In Coursework. In This Book There Is No Disclosure With Many Of The Data Sets What Type Of Investigation Should Be Made Or What Assumptions Are To Be Used. most Reliability Practitioners Will Be Employed Where Personal Interaction Between Disciplines Is A Necessity. A Section Is Included On Communication Skills To Facilitate Model Selection And Formulation Based On Verifiable Assumptions, Rather Than Favorable Conclusions. However, Whether The Answer Is Right Can Never Be Ascertained. past And Current Applications Of Stochastic Modeling To Life-length Can Only Be A Guide For Future Adaptations Under Different Conditions, With New Materials In Unknown Usages. This Book Unifies The Study Of Cumulative-damage Distributions, Namely, Wald And Tweedie (i.e., Inverse-gaussian And Its Reciprocal) With Fatigue-life. These Distributions Are Most Useful When The Coefficient-of-variation Is More Appropriate Than Is The Variance As A Measure Of Dispersion. It Is Shown, Uniquely, That The Same Hyperbolic-sine Transformation Of Each Life Length Variate Has A Chi-square One-df Distribution. This Property Is Useful In The Sample Statistics. These Ihra Distributions Realistically Model Life-length, Strength Or Duration Of Load Under Linear Cumulative Damage And Can Be Combined As Approximations In Non-linear Situations. sam C. Saunders Has Served As A Research Engineer For 17 Years At The Boeing Scientific Research Laboratories, 20 Years As A Consultant To The Advisory Committee For Nuclear Safeguards, 10 Years As A Consultant To Nist, Was A Principal In The Consulting Firms Mathematical Analysis Research Corporation And Scientific Consulting Service; And Was For 26 Years A Professor Of Applied Mathematics/statistics At Washington State University. He Is A Fellow Of The American Statistical Association And A Former Editor Of Technometrics. Springer 0387325220 1 Reliability, Life Testing and the Prediction of Service Lives 4 Copyright Page 5 Preface 6 Acknowledgements 8 Vörtrekkers 8 Glossary 9 Admonitions 10 Table of Contents 12 CHAPTER 1 Requisites 16 1.1. Why Reliability Is Important 16 1.2. Valuable Concepts 18 1.2.1. Concepts from Probability 18 1.2.2. Concepts from Statistics 21 CHAPTER 2 Elements of Reliability 25 2.1. Properties of Life Distributions 25 2.2. Useful Parametric Life Distributions 29 2.2.1. The Epstein (Exponential) Distribution 29 2.2.2. The Gamma Distribution 30 2.2.3. The Pareto Distribution 30 2.2.4. The Gaussian or Normal Distribution 31 2.2.5. Transformations to Normality 32 The Truncated Normal Distribution 32 The Log-Normal Distribution 33 The Xi-Normal Family 34 2.2.6. The Fatigue-Life Distribution 35 2.2.7. The Inverse-Gaussian Distribution 35 2.2.8. The Extreme-Value Distribution of Minima 36 2.2.9. Some Other Distributions 37 CHAPTER 3 Partitions and Selection 41 3.1. Binomial Coefficients and Sterling Numbers 41 3.1.1. Lagrange Coefficients 43 3.2. Lotteries and Coupon Collecting 45 3.2.1. Lotteries 45 3.2.2. Coupon Collecting 46 3.3. Occupancy and Allocations 49 3.3.1. Occupancy 49 Multiple Occupancy 51 3.3.2. Allocations 53 3.4. Related Concepts 54 3.4.1. The Sum of Epstein Waiting Times 54 3.4.2. Interpolation and Numerical Integration 55 CHAPTER 4 Coherent Systems 59 4.1. Functional Representation 59 4.2. Event-Tree Depiction 65 4.2.1. Associated Random Variables 66 4.3. Evaluation of Reliability 68 4.3.1. System Life 70 4.4. Use of Association to Bound Reliability 75 4.5. Shape of the Reliability Function 78 4.6. Diagnostics and Importance of System Components 81 4.6.1. Importance 81 4.6.2. Diagnostics Using Reliability 81 4.7. Hazard Rates and Pόlya Frequency Functions 83 4.8. Closure Properties 84 4.8.1. Further Closure Properties 86 CHAPTER 5 Applicable Life Distributions 90 5.1. The Gaussian or Normal Distribution 90 5.2. Epstein's Distribution 92 5.2.1. The Erlang-k Distribution 93 5.3. The Galton and Fatigue-Life Distributions 93 5.3.1. The Log-Normal Distribution 93 5.3.2. The Fatigue-Life Distribution 94 5.4. Discovery and Rediscovery 95 5.5. Extreme Value Theory and Association 97 5.5.1. Gumbel's Theory 97 5.5.2. Maximum Loads and Association 99 CHAPTER 6 Philosophy, Science, and Sense 104 6.1. Likelihood without Priors 104 6.2. Likelihood for Complete Samples 107 6.3. Properties of the Likelihood 109 6.3.1. The Likelihood Depends upon the Model 109 6.3.2. Relative Likelihoods Are Not Probabilities on Θ 110 6.3.3. Likelihoods Invariant under Transformations 111 6.3.4. Likelihoods on Simple Parameter Spaces 111 6.3.5. Bayes' Theorem and Its Application 113 6.4. Types of Censoring of Data 116 6.4.1. Estimation for Type I Censoring 117 6.4.2. Estimation for Type II Censoring 117 6.4.3. Estimation for Type III Random Censoring 118 6.4.4. Transformation to the Standard Weibull 119 6.5. Generation of Ordered Observations 120 6.6. A Parametric Model of Censoring 123 6.7. The Empirical Cumulative Distribution 126 CHAPTER 7 Nonparametric Life Estimators 129 7.1. The Empiric Survival Distribution 129 7.1.1. Life-Table Methods 129 The Reduced Sample Method 130 The Actuarial Method 130 7.1.2. The Kaplan-Meier Estimator 130 7.2. Expectation and Bias of the K-M Estimator 132 Proportional Hazards 135 7.3. The Variance and Mean-Square Error 137 7.4. The Nelson-Aalen Estimator 139 7.4.1. Extensions and Generalizations 140 CHAPTER 8 Weibull Analysis 143 8.1. Distribution of Failure Times for Systems 143 8.2. Estimation for the Weibull Distribution 143 8.2.1. Right-Censored Estimation 144 8.2.2. Left-Censored Estimation 144 8.3. Competing Risks 145 8.3.1. The Bathtub-Shaped Hazard 145 8.4. Analysis of Censored Data 146 8.4.1. Estimation under Independent Competing Risks 146 8.4.2. Observing Both Time and Cause of Failure 147 8.4.3. Estimation with Dependent Failure Modes 149 8.4.4. Estimation under Random Censoring on Both Sides 150 8.4.5. Censoring for the Reciprocal Weibull 152 8.5. Change Points and Multiple Failure Mechanisms 154 8.5.1. A Known Change Point 155 8.5.2. A Change Point at an Unknown Location 159 8.5.3. Conclusions 161 CHAPTER 9 Examine Data, Diagnose and Consult 163 9.1. Scientific Idealism 163 9.2. Consultation and Diagnosis 164 9.3. Datasets in Service-Life Prediction 166 9.4. Data, Consulting, and Modeling 172 CHAPTER 10 Cumulative Damage Distributions 175 10.1. The Past as Prologue 175 10.2. The Fatigue-Life Distribution 177 10.3. The Mixed Class of Cumulative Damage Distributions 179 10.4. Elementary Derivation of Means and Variances 181 10.5. Behavior of the Hazard Rate 183 10.6. Mixed Variate Relationships 187 10.7. Estimation for Wald's Distributions 191 10.7.1. Estimation for Complete Samples 191 Estimation of α When β Is Known 192 Estimation of β When α Is Known 192 Unbised Estimation 193 10.7.2. Estimation for Incomplete Wald Samples 195 10.8. Estimation for the FL-Distribution 197 10.8.1. Complete Samples 197 10.8.2. Incomplete Samples of Fatigue-life Distribution 199 10.9. Estimation for Tweedie's Distribution 202 10.10. Cases of Misidentification 204 10.10.1. When the FL-Distribution Is Unknown 204 10.10.2. When the CD-Distributions Are Unknown 204 10.10.3. Weibull Distribution Contrasted with the FL-Distribution 205 10.10.4. Galton Distribution Mistaken for FL-Distribution 206 CHAPTER 11 Analysis of Dispersion 209 11.1. Applicability 209 11.2. Schrödinger's Distribution 210 11.3. Sample Distributions under Consonance 210 11.3.1. And Student's Distribution? 219 11.4. Classifications for Dispersion Analysis 221 11.4.1. A Single Classification 222 11.4.2. A Two-Way Classification for Multiplicative Effects 223 No Row or Column Effects 223 No Column Effects 224 No Row Effects 226 When Does Consonance Occur? 226 CHAPTER 12 Damage Processes 229 12.1. The Poisson Process 229 12.1.1. The Superposition of Poisson Processes 231 12.1.2. The Decomposition of Poisson Processes 231 12.2. Damage Due to Intermittant Shocks 231 12.3. Renewal Processes 234 12.3.1. Renewal Function for the Wald Distribution 236 12.3.2. Negligible Replacement Times for Units in Service 238 12.3.3. Tauberian Theorems for the Laplace Transform 238 12.4. Shock Models with Varying Intensity 239 12.4.1. The Marshall-Olkin Distribution 240 12.4.2. The Bivariate Poisson 242 12.5. Stationary Renewal Processes 242 12.6. The Miner-Palmgren Rule and Additive Damage 244 12.6.1. Miner's Rule as an Expectation 245 12.6.2. How Applicable Is This Theory? 246 12.7. Other Cumulative Damage Processes 247 12.7.1. Deterioration of Polymer Coatings 247 12.7.2. Varying Duty Cycles 248 12.8. When Linear Cumulative Damage Fails 250 12.8.1. Load-Order Effects in Crack Propagation 251 CHAPTER 13 Service Life of Structures 255 13.1. Wear under Spectral Loading 255 13.2. Multivariate Fatigue Life 256 13.2.1. Two-Component Load Sharing 257 13.2.2. The Multivariate Fatigue-Life Distribution 258 13.3. Correlations between Component Damage 263 13.3.1. Covariance and Association 264 13.4. Implementation 268 13.4.1. Estimation for Small Censored Samples 269 13.4.2. Relating Cumulative-Damage Parameters to the Exposure 270 CHAPTER 14 Strength and Durability 273 14.1. Range of Applicability 273 14.1.1. Introduction 273 14.1.2. Reliability Analysis of Strength 274 Static Strength for a Column 274 14.1.3. Strength of an Airframe Subsystem 276 14.2. Accelerated Tests for Strength 277 14.2.1. Determination of the Part of Least Accord 279 14.3. Danger of Extrapolation from Tests 281 14.3.1. Relating Parameters to the Exposure 283 The Pagett Models Using the Wald Distribution 283 14.4. Fracture Mechanics and Stochastic Damage 285 CHAPTER 15 Maintenance of Systems 288 15.1. Introduction 288 15.2. Availability 288 15.2.1. Application of Tauberian Properties 290 15.2.2. System Availability 291 Systems with Spares 292 15.3. Age Replacement with Renewal 292 15.3.1. A Single Machine with Repair 294 15.4. The Inversion of Transforms 296 15.5. Problems in Scheduled Maintenance 299 15.5.1. A Problem with Unscheduled Fleet Maintenance 300 15.5.2. A Problem with Scheduled Fleet Maintenance 301 CHAPTER 16 Mathematical Appendix 304 16.1. Integration 304 16.1.1. Stieltjes Integrals 304 16.2. Probability and Measure 306 16.3. Distribution Transforms 308 16.4. A Compendium of Discrete Distributions 312 16.5. A Compendium of Continuous Distributions 313 Bibliography 314 Index 320 ISBN-13:,9780387325224 0387325220......Page 1 Reliability, Life Testing and the Prediction of Service Lives......Page 4 Copyright Page......Page 5 Preface......Page 6 Vörtrekkers......Page 8 Glossary......Page 9 Admonitions......Page 10 Table of Contents......Page 12 1.1. Why Reliability Is Important......Page 16 1.2.1. Concepts from Probability......Page 18 1.2.2. Concepts from Statistics......Page 21 2.1. Properties of Life Distributions......Page 25 2.2.1. The Epstein (Exponential) Distribution......Page 29 2.2.3. The Pareto Distribution......Page 30 2.2.4. The Gaussian or Normal Distribution......Page 31 The Truncated Normal Distribution......Page 32 The Log-Normal Distribution......Page 33 The Xi-Normal Family......Page 34 2.2.7. The Inverse-Gaussian Distribution......Page 35 2.2.8. The Extreme-Value Distribution of Minima......Page 36 2.2.9. Some Other Distributions......Page 37 3.1. Binomial Coefficients and Sterling Numbers......Page 41 3.1.1. Lagrange Coefficients......Page 43 3.2.1. Lotteries......Page 45 3.2.2. Coupon Collecting......Page 46 3.3.1. Occupancy......Page 49 Multiple Occupancy......Page 51 3.3.2. Allocations......Page 53 3.4.1. The Sum of Epstein Waiting Times......Page 54 3.4.2. Interpolation and Numerical Integration......Page 55 4.1. Functional Representation......Page 59 4.2. Event-Tree Depiction......Page 65 4.2.1. Associated Random Variables......Page 66 4.3. Evaluation of Reliability......Page 68 4.3.1. System Life......Page 70 4.4. Use of Association to Bound Reliability......Page 75 4.5. Shape of the Reliability Function......Page 78 4.6.2. Diagnostics Using Reliability......Page 81 4.7. Hazard Rates and Pόlya Frequency Functions......Page 83 4.8. Closure Properties......Page 84 4.8.1. Further Closure Properties......Page 86 5.1. The Gaussian or Normal Distribution......Page 90 5.2. Epstein's Distribution......Page 92 5.3.1. The Log-Normal Distribution......Page 93 5.3.2. The Fatigue-Life Distribution......Page 94 5.4. Discovery and Rediscovery......Page 95 5.5.1. Gumbel's Theory......Page 97 5.5.2. Maximum Loads and Association......Page 99 6.1. Likelihood without Priors......Page 104 6.2. Likelihood for Complete Samples......Page 107 6.3.1. The Likelihood Depends upon the Model......Page 109 6.3.2. Relative Likelihoods Are Not Probabilities on Θ......Page 110 6.3.4. Likelihoods on Simple Parameter Spaces......Page 111 6.3.5. Bayes' Theorem and Its Application......Page 113 6.4. Types of Censoring of Data......Page 116 6.4.2. Estimation for Type II Censoring......Page 117 6.4.3. Estimation for Type III Random Censoring......Page 118 6.4.4. Transformation to the Standard Weibull......Page 119 6.5. Generation of Ordered Observations......Page 120 6.6. A Parametric Model of Censoring......Page 123 6.7. The Empirical Cumulative Distribution......Page 126 7.1.1. Life-Table Methods......Page 129 7.1.2. The Kaplan-Meier Estimator......Page 130 7.2. Expectation and Bias of the K-M Estimator......Page 132 Proportional Hazards......Page 135 7.3. The Variance and Mean-Square Error......Page 137 7.4. The Nelson-Aalen Estimator......Page 139 7.4.1. Extensions and Generalizations......Page 140 8.2. Estimation for the Weibull Distribution......Page 143 8.2.2. Left-Censored Estimation......Page 144 8.3.1. The Bathtub-Shaped Hazard......Page 145 8.4.1. Estimation under Independent Competing Risks......Page 146 8.4.2. Observing Both Time and Cause of Failure......Page 147 8.4.3. Estimation with Dependent Failure Modes......Page 149 8.4.4. Estimation under Random Censoring on Both Sides......Page 150 8.4.5. Censoring for the Reciprocal Weibull......Page 152 8.5. Change Points and Multiple Failure Mechanisms......Page 154 8.5.1. A Known Change Point......Page 155 8.5.2. A Change Point at an Unknown Location......Page 159 8.5.3. Conclusions......Page 161 9.1. Scientific Idealism......Page 163 9.2. Consultation and Diagnosis......Page 164 9.3. Datasets in Service-Life Prediction......Page 166 9.4. Data, Consulting, and Modeling......Page 172 10.1. The Past as Prologue......Page 175 10.2. The Fatigue-Life Distribution......Page 177 10.3. The Mixed Class of Cumulative Damage Distributions......Page 179 10.4. Elementary Derivation of Means and Variances......Page 181 10.5. Behavior of the Hazard Rate......Page 183 10.6. Mixed Variate Relationships......Page 187 10.7.1. Estimation for Complete Samples......Page 191 Estimation of β When α Is Known......Page 192 Unbised Estimation......Page 193 10.7.2. Estimation for Incomplete Wald Samples......Page 195 10.8.1. Complete Samples......Page 197 10.8.2. Incomplete Samples of Fatigue-life Distribution......Page 199 10.9. Estimation for Tweedie's Distribution......Page 202 10.10.2. When the CD-Distributions Are Unknown......Page 204 10.10.3. Weibull Distribution Contrasted with the FL-Distribution......Page 205 10.10.4. Galton Distribution Mistaken for FL-Distribution......Page 206 11.1. Applicability......Page 209 11.3. Sample Distributions under Consonance......Page 210 11.3.1. And Student's Distribution?......Page 219 11.4. Classifications for Dispersion Analysis......Page 221 11.4.1. A Single Classification......Page 222 No Row or Column Effects......Page 223 No Column Effects......Page 224 When Does Consonance Occur?......Page 226 12.1. The Poisson Process......Page 229 12.2. Damage Due to Intermittant Shocks......Page 231 12.3. Renewal Processes......Page 234 12.3.1. Renewal Function for the Wald Distribution......Page 236 12.3.3. Tauberian Theorems for the Laplace Transform......Page 238 12.4. Shock Models with Varying Intensity......Page 239 12.4.1. The Marshall-Olkin Distribution......Page 240 12.5. Stationary Renewal Processes......Page 242 12.6. The Miner-Palmgren Rule and Additive Damage......Page 244 12.6.1. Miner's Rule as an Expectation......Page 245 12.6.2. How Applicable Is This Theory?......Page 246 12.7.1. Deterioration of Polymer Coatings......Page 247 12.7.2. Varying Duty Cycles......Page 248 12.8. When Linear Cumulative Damage Fails......Page 250 12.8.1. Load-Order Effects in Crack Propagation......Page 251 13.1. Wear under Spectral Loading......Page 255 13.2. Multivariate Fatigue Life......Page 256 13.2.1. Two-Component Load Sharing......Page 257 13.2.2. The Multivariate Fatigue-Life Distribution......Page 258 13.3. Correlations between Component Damage......Page 263 13.3.1. Covariance and Association......Page 264 13.4. Implementation......Page 268 13.4.1. Estimation for Small Censored Samples......Page 269 13.4.2. Relating Cumulative-Damage Parameters to the Exposure......Page 270 14.1.1. Introduction......Page 273 Static Strength for a Column......Page 274 14.1.3. Strength of an Airframe Subsystem......Page 276 14.2. Accelerated Tests for Strength......Page 277 14.2.1. Determination of the Part of Least Accord......Page 279 14.3. Danger of Extrapolation from Tests......Page 281 The Pagett Models Using the Wald Distribution......Page 283 14.4. Fracture Mechanics and Stochastic Damage......Page 285 15.2. Availability......Page 288 15.2.1. Application of Tauberian Properties......Page 290 15.2.2. System Availability......Page 291 15.3. Age Replacement with Renewal......Page 292 15.3.1. A Single Machine with Repair......Page 294 15.4. The Inversion of Transforms......Page 296 15.5. Problems in Scheduled Maintenance......Page 299 15.5.1. A Problem with Unscheduled Fleet Maintenance......Page 300 15.5.2. A Problem with Scheduled Fleet Maintenance......Page 301 16.1.1. Stieltjes Integrals......Page 304 16.2. Probability and Measure......Page 306 16.3. Distribution Transforms......Page 308 16.4. A Compendium of Discrete Distributions......Page 312 16.5. A Compendium of Continuous Distributions......Page 313 Bibliography......Page 314 Index......Page 320 This book is intended for students and practitioners who have had a calculus-based statistics course and who have an interest in safety considerations such as reliability, strength, and duration-of-load or service life. Many persons studying statistical science will be employed professionally where the problems encountered are obscure, what should be analyzed is not clear, the appropriate assumptions are equivocal, and data are scant. Yet tutorial problems of this nature are virtually never encountered in coursework. In this book there is no disclosure with many of the data sets what type of investigation should be made or what assumptions are to be used. Most reliability practitioners will be employed where personal interaction between disciplines is a necessity. A section is included on communication skills to facilitate model selection and formulation based on verifiable assumptions, rather than favorable conclusions. However, whether the answer is "right" can never be ascertained. Past and current applications of stochastic modeling to life-length can only be a guide for future adaptations under different conditions, with new materials in unknown usages. This book unifies the study of cumulative-damage distributions, namely, Wald and Tweedie (i.e., inverse-Gaussian and its reciprocal) with "fatigue-life." These distributions are most useful when the coefficient-of-variation is more appropriate than is the variance as a measure of dispersion. It is shown, uniquely, that the same hyperbolic-sine transformation of each life length variate has a Chi-square one-df distribution. This property is useful in the sample statistics. These IHRA distributions realistically model life-length, strength or duration of load under linear cumulative damage and can be combined as approximations in non-linear situations. Sam C. Saunders has served as a research engineer for 17 years at the Boeing Scientific Research Laboratories, 20 years as a consultant to the Advisory Committee for Nuclear Safeguards, 10 years as a consultant to NIST, was a principal in the consulting firms Mathematical Analysis Research Corporation and Scientific Consulting Service; and was for 26 years a professor of Applied Mathematics/Statistics at Washington State University. He is a Fellow of the American Statistical Association and a former editor of Technometrics The prerequisite for reading this text is a calculus based course in Probability and Mathematical Statistics, along with the usual curricularmathematical requi- ments for every science major. For graduate students from disciplines other than mathematical sciences much advantage, viz., both insight and mathematical - turity, is gained by having had experience quantifying the assurance for safety of structures, operability of systems or health of persons. It is presumed that each student will have some familiarity with Mathematica or Maple or better yet also have available some survival analysis software such as S Plus or R, to handle the computations with the data sets. This material has been selected under the conviction that the most practical aid any investigator can have is a good theory. The course is intended for p- sons who will, during their professional life, be concerned with the'theoretical'aspects of applied science. This implies consulting with industrial mathema- cians/statisticians'lead engineers in various fields, physcists, chemists, material scientists and other technical specialists who are collaborating to solve some d- ficult technological/scientific problem. Accordingly, there are sections devoted to the deportment of applied mathematicians during consulting. This corresponds to the'bedside manner'of physicians and is a important aspect of professionalism.