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

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. Title Page Copyright Page Preface Acknowledgements Vörtrekkers Glossary Admonitions Table of Contents CHAPTER 1 Requisites 1.1. Why Reliability Is Important 1.2. Valuable Concepts 1.2.1. Concepts from Probability 1.2.2. Concepts from Statistics CHAPTER 2 Elements of Reliability 2.1. Properties of Life Distributions 2.2. Useful Parametric Life Distributions 2.2.1. The Epstein (Exponential) Distribution 2.2.2. The Gamma Distribution 2.2.3. The Pareto Distribution 2.2.4. The Gaussian or Normal Distribution 2.2.5. Transformations to Normality The Truncated Normal Distribution The Log-Normal Distribution The Xi-Normal Family 2.2.6. The Fatigue-Life Distribution 2.2.7. The Inverse-Gaussian Distribution 2.2.8. The Extreme-Value Distribution of Minima 2.2.9. Some Other Distributions CHAPTER 3 Partitions and Selection 3.1. Binomial Coefficients and Sterling Numbers 3.1.1. Lagrange Coefficients 3.2. Lotteries and Coupon Collecting 3.2.1. Lotteries 3.2.2. Coupon Collecting 3.3. Occupancy and Allocations 3.3.1. Occupancy Multiple Occupancy 3.3.2. Allocations 3.4. Related Concepts 3.4.1. The Sum of Epstein Waiting Times 3.4.2. Interpolation and Numerical Integration CHAPTER 4 Coherent Systems 4.1. Functional Representation 4.2. Event-Tree Depiction 4.2.1. Associated Random Variables 4.3. Evaluation of Reliability 4.3.1. System Life 4.4. Use of Association to Bound Reliability 4.5. Shape of the Reliability Function 4.6. Diagnostics and Importance of System Components 4.6.1. Importance 4.6.2. Diagnostics Using Reliability 4.7. Hazard Rates and Pόlya Frequency Functions 4.8. Closure Properties 4.8.1. Further Closure Properties CHAPTER 5 Applicable Life Distributions 5.1. The Gaussian or Normal Distribution 5.2. Epstein's Distribution 5.2.1. The Erlang-k Distribution 5.3. The Galton and Fatigue-Life Distributions 5.3.1. The Log-Normal Distribution 5.3.2. The Fatigue-Life Distribution 5.4. Discovery and Rediscovery 5.5. Extreme Value Theory and Association 5.5.1. Gumbel's Theory 5.5.2. Maximum Loads and Association CHAPTER 6 Philosophy, Science, and Sense 6.1. Likelihood without Priors 6.2. Likelihood for Complete Samples 6.3. Properties of the Likelihood 6.3.1. The Likelihood Depends upon the Model 6.3.2. Relative Likelihoods Are Not Probabilities on Θ 6.3.3. Likelihoods Invariant under Transformations 6.3.4. Likelihoods on Simple Parameter Spaces 6.3.5. Bayes' Theorem and Its Application 6.4. Types of Censoring of Data 6.4.1. Estimation for Type I Censoring 6.4.2. Estimation for Type II Censoring 6.4.3. Estimation for Type III Random Censoring 6.4.4. Transformation to the Standard Weibull 6.5. Generation of Ordered Observations 6.6. A Parametric Model of Censoring 6.7. The Empirical Cumulative Distribution CHAPTER 7 Nonparametric Life Estimators 7.1. The Empiric Survival Distribution 7.1.1. Life-Table Methods The Reduced Sample Method The Actuarial Method 7.1.2. The Kaplan-Meier Estimator 7.2. Expectation and Bias of the K-M Estimator Proportional Hazards 7.3. The Variance and Mean-Square Error 7.4. The Nelson-Aalen Estimator 7.4.1. Extensions and Generalizations CHAPTER 8 Weibull Analysis 8.1. Distribution of Failure Times for Systems 8.2. Estimation for the Weibull Distribution 8.2.1. Right-Censored Estimation 8.2.2. Left-Censored Estimation 8.3. Competing Risks 8.3.1. The Bathtub-Shaped Hazard 8.4. Analysis of Censored Data 8.4.1. Estimation under Independent Competing Risks 8.4.2. Observing Both Time and Cause of Failure 8.4.3. Estimation with Dependent Failure Modes 8.4.4. Estimation under Random Censoring on Both Sides 8.4.5. Censoring for the Reciprocal Weibull 8.5. Change Points and Multiple Failure Mechanisms 8.5.1. A Known Change Point 8.5.2. A Change Point at an Unknown Location 8.5.3. Conclusions CHAPTER 9 Examine Data, Diagnose and Consult 9.1. Scientific Idealism 9.2. Consultation and Diagnosis 9.3. Datasets in Service-Life Prediction 9.4. Data, Consulting, and Modeling CHAPTER 10 Cumulative Damage Distributions 10.1. The Past as Prologue 10.2. The Fatigue-Life Distribution 10.3. The Mixed Class of Cumulative Damage Distributions 10.4. Elementary Derivation of Means and Variances 10.5. Behavior of the Hazard Rate 10.6. Mixed Variate Relationships 10.7. Estimation for Wald's Distributions 10.7.1. Estimation for Complete Samples Estimation of α When β Is Known Estimation of β When α Is Known Unbised Estimation 10.7.2. Estimation for Incomplete Wald Samples 10.8. Estimation for the FL-Distribution 10.8.1. Complete Samples 10.8.2. Incomplete Samples of Fatigue-life Distribution 10.9. Estimation for Tweedie's Distribution 10.10. Cases of Misidentification 10.10.1. When the FL-Distribution Is Unknown 10.10.2. When the CD-Distributions Are Unknown 10.10.3. Weibull Distribution Contrasted with the FL-Distribution 10.10.4. Galton Distribution Mistaken for FL-Distribution CHAPTER 11 Analysis of Dispersion 11.1. Applicability 11.2. Schrödinger's Distribution 11.3. Sample Distributions under Consonance 11.3.1. And Student's Distribution? 11.4. Classifications for Dispersion Analysis 11.4.1. A Single Classification 11.4.2. A Two-Way Classification for Multiplicative Effects No Row or Column Effects No Column Effects No Row Effects When Does Consonance Occur? CHAPTER 12 Damage Processes 12.1. The Poisson Process 12.1.1. The Superposition of Poisson Processes 12.1.2. The Decomposition of Poisson Processes 12.2. Damage Due to Intermittant Shocks 12.3. Renewal Processes 12.3.1. Renewal Function for the Wald Distribution 12.3.2. Negligible Replacement Times for Units in Service 12.3.3. Tauberian Theorems for the Laplace Transform 12.4. Shock Models with Varying Intensity 12.4.1. The Marshall-Olkin Distribution 12.4.2. The Bivariate Poisson 12.5. Stationary Renewal Processes 12.6. The Miner-Palmgren Rule and Additive Damage 12.6.1. Miner's Rule as an Expectation 12.6.2. How Applicable Is This Theory? 12.7. Other Cumulative Damage Processes 12.7.1. Deterioration of Polymer Coatings 12.7.2. Varying Duty Cycles 12.8. When Linear Cumulative Damage Fails 12.8.1. Load-Order Effects in Crack Propagation CHAPTER 13 Service Life of Structures 13.1. Wear under Spectral Loading 13.2. Multivariate Fatigue Life 13.2.1. Two-Component Load Sharing 13.2.2. The Multivariate Fatigue-Life Distribution 13.3. Correlations between Component Damage 13.3.1. Covariance and Association 13.4. Implementation 13.4.1. Estimation for Small Censored Samples 13.4.2. Relating Cumulative-Damage Parameters to the Exposure CHAPTER 14 Strength and Durability 14.1. Range of Applicability 14.1.1. Introduction 14.1.2. Reliability Analysis of Strength Static Strength for a Column 14.1.3. Strength of an Airframe Subsystem 14.2. Accelerated Tests for Strength 14.2.1. Determination of the Part of Least Accord 14.3. Danger of Extrapolation from Tests 14.3.1. Relating Parameters to the Exposure The Pagett Models Using the Wald Distribution 14.4. Fracture Mechanics and Stochastic Damage CHAPTER 15 Maintenance of Systems 15.1. Introduction 15.2. Availability 15.2.1. Application of Tauberian Properties 15.2.2. System Availability Systems with Spares 15.3. Age Replacement with Renewal 15.3.1. A Single Machine with Repair 15.4. The Inversion of Transforms 15.5. Problems in Scheduled Maintenance 15.5.1. A Problem with Unscheduled Fleet Maintenance 15.5.2. A Problem with Scheduled Fleet Maintenance CHAPTER 16 Mathematical Appendix 16.1. Integration 16.1.1. Stieltjes Integrals 16.2. Probability and Measure 16.3. Distribution Transforms 16.4. A Compendium of Discrete Distributions 16.5. A Compendium of Continuous Distributions Bibliography Index

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.

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