Reliability and Risk

Nozer D. Singpurwalla is the author of Reliability and Risk: A Bayesian Perspective, published by Wiley.

"The book is written by an expert in reliability analysis and it is a very valuable source of information for mathematical models for reliability problems ... An extensive bibliography concludes the book." (Stat Papers, 2011) "As the author mentions in his preface, the book can be read in several different ways, as a text for a graduate level course on reliability or as a source book for “information and open problems." This book has been a joy to read for this reviewer." (International Statistical Review, August 2008)"Singpurwalla seems to be at his best in probabilistic modeling of reality. He has written what must be one of the first books reliability written from a subjective, Bayesian point of view." (International Statistical Review, August 2008)"The material of this book will be most profitable for practitioners and researchers in reliability and survivability, who will greatly appreciate it as a source of information and open problems." (Mathematical Reviews, 2008h)"This is a very interesting, provocative, and worthwhile book." (Biometrics, June 2008)"What I liked most about this book, however, is the way it blends interesting technical material with foundational discussion about the nature of uncertainty." (Biometrics, June 2008)"The investigation of the theoretical models under consideration in the book is first class…" (Law, Probability and Risk Advance Access, September 2007)"I feel that I have learned an effective plotting technique from these plots…" (Technometrics, February 2008)"…a cornucopia of probability models and inference methods for different problems…[that] serve as a rich taxonomy that statisticians can use to fit models…works as both an educational tool and as a reference." (MAA Reviews, March 6, 2007)

• Preface xiiiAcknowledgements xv1 Introduction and Overview 11.1 Preamble: What do ‘Reliability’, ‘Risk’ and ‘Robustness’ Mean? 11.2 Objectives and Prospective Readership 31.3 Reliability, Risk and Survival: State-of-the-Art 31.4 Risk Management: A Motivation for Risk Analysis 41.5 Books on Reliability, Risk and Survival Analysis 61.6 Overview of the Book 72 The Quantification of Uncertainty 92.1 Uncertain Quantities and Uncertain Events: Their Definition and Codification 92.2 Probability: A Satisfactory Way to Quantify Uncertainty 102.2.1 The Rules of Probability 112.2.2 Justifying the Rules of Probability 122.3 Overview of the Different Interpretations of Probability 132.3.1 A Brief History of Probability 142.3.2 The Different Kinds of Probability 162.4 Extending the Rules of Probability: Law of Total Probability and Bayes’ Law 192.4.1 Marginalization 202.4.2 The Law of Total Probability 202.4.3 Bayes’ Law: The Incorporation of Evidence and the Likelihood 202.5 The Bayesian Paradigm: A Prescription for Reliability, Risk and Survival Analysis 222.6 Probability Models, Parameters, Inference and Prediction 232.6.1 The Genesis of Probability Models and Their Parameters 242.6.2 Statistical Inference and Probabilistic Prediction 262.7 Testing Hypotheses: Posterior Odds and Bayes Factors 272.7.1 Bayes Factors: Weight of Evidence and Change in Odds 282.7.2 Uses of the Bayes Factor 302.7.3 Alternatives to Bayes Factors 312.8 Utility as Probability and Maximization of Expected Utility 322.8.1 Utility as a Probability 322.8.2 Maximization of Expected Utility 332.8.3 Attitudes to Risk: The Utility of Money 332.9 Decision Trees and Influence Diagrams for Risk Analysis 342.9.1 The Decision Tree 342.9.2 The Influence Diagram 353 Exchangeability and Indifference 453.1 Introduction to Exchangeability: de Finetti’s Theorem 453.1.1 Motivation for the Judgment of Exchangeability 463.1.2 Relationship between Independence and Exchangeability 463.1.3 de Finetti’s Representation Theorem for Zero-one Exchangeable Sequences 483.1.4 Exchangeable Sequences and the Law of Large Numbers 493.2 de Finetti-style Theorems for Infinite Sequences of Non-binary Random Quantities 503.2.1 Sufficiency and Indifference in Zero-one Exchangeable Sequences 513.2.2 Invariance Conditions Leading to Mixtures of Other Distributions 513.3 Error Bounds on de Finetti-style Results for Finite Sequences of Random Quantities 553.3.1 Bounds for Finitely Extendable Zero-one Random Quantities 553.3.2 Bounds for Finitely Extendable Non-binary Random Quantities 564 Stochastic Models of Failure 594.1 Introduction 594.2 Preliminaries: Univariate, Multivariate and Multi-indexed Distribution Functions 594.3 The Predictive Failure Rate Function of a Univariate Probability Distribution 624.3.1 The Case of Discontinuity 654.4 Interpretation and Uses of the Failure Rate Function – the Model Failure Rate 664.4.1 The True Failure Rate: Does it Exist? 694.4.2 Decreasing Failure Rates, Reliability Growth, Burn-in and the Bathtub Curve 694.4.3 The Retrospective (or Reversed) Failure Rate 744.5 Multivariate Analogues of the Failure Rate Function 764.5.1 The Hazard Gradient 764.5.2 The Multivariate Failure Rate Function 774.5.3 The Conditional Failure Rate Functions 784.6 The Hazard Potential of Items and Individuals 794.6.1 Hazard Potentials and Dependent Lifelengths 814.6.2 The Hazard Gradient and Conditional Hazard Potentials 834.7 Probability Models for Interdependent Lifelengths 854.7.1 Preliminaries: Bivariate Distributions 854.7.2 The Bivariate Exponential Distributions of Gumbel 894.7.3 Freund’s Bivariate Exponential Distribution 914.7.4 The Bivariate Exponential of Marshall and Olkin 934.7.5 The Bivariate Pareto as a Failure Model 1074.7.6 A Bivariate Exponential Induced by a Shot-noise Process 1104.7.7 A Bivariate Exponential Induced by a Bivariate Pareto’s Copula 1154.7.8 Other Specialized Bivariate Distributions 1154.8 Causality and Models for Cascading Failures 1174.8.1 Probabilistic Causality and Causal Failures 1174.8.2 Cascading and Models of Cascading Failures 1184.9 Failure Distributions with Multiple Scales 1204.9.1 Model Development 1204.9.2 A Failure Model Indexed by Two Scales 1235 Parametric Failure Data Analysis 1255.1 Introduction and Perspective 1255.2 Assessing Predictive Distributions in the Absence of Data 1275.2.1 The Exponential as a Chance Distribution 1275.2.2 The Weibull (and Gamma) as a Chance Distribution 1285.2.3 The Bernoulli as a Chance Distribution 1295.2.4 The Poisson as a Chance Distribution 1335.2.5 The Generalized Gamma as a Chance Distribution 1355.2.6 The Inverse Gaussian as a Chance Distribution 1365.3 Prior Distributions in Chance Distributions 1365.3.1 Eliciting Prior Distributions via Expert Testimonies 1375.3.2 Using Objective (or Default) Priors 1415.4 Predictive Distributions Incorporating Failure Data 1445.4.1 Design Strategies for Industrial Life-testing 1455.4.2 Stopping Rules: Non-informative and Informative 1475.4.3 The Total Time on Test 1495.4.4 Exponential Life-testing Procedures 1505.4.5 Weibull Life-testing Procedures 1555.4.6 Life-testing Under the Generalized Gamma and the Inverse Gaussian 1565.4.7 Bernoulli Life-testing Procedures 1575.4.8 Life-testing and Inference Under the BVE 1595.5 Information from Life-tests: Learning from Data 1615.5.1 Preliminaries: Entropy and Information 1615.5.2 Learning for Inference from Life-test Data: Testing for Confidence 1645.5.3 Life-testing for Decision Making: Acceptance Sampling 1665.6 Optimal Testing: Design of Life-testing Experiments 1705.7 Adversarial Life-testing and Acceptance Sampling 1735.8 Accelerated Life-testing and Dose–response Experiments 1755.8.1 Formulating Accelerated Life-testing Problems 1755.8.2 The Kalman Filter Model for Prediction and Smoothing 1775.8.3 Inference from Accelerated Tests Using the Kalman Filter 1795.8.4 Designing Accelerated Life-testing Experiments 1836 Composite Reliability: Signatures 1876.1 Introduction: Hierarchical Models 1876.2 ‘Composite Reliability’: Partial Exchangeability 1886.2.1 Simulating Exchangeable and Partially Exchangeable Sequences 1896.2.2 The Composite Reliability of Ultra-reliable Units 1906.2.3 Assessing Reliability and Composite Reliability 1926.3 Signature Analysis and Signatures as Covariates 1936.3.1 Assessing the Power Spectrum via a Regression Model 1956.3.2 Bayesian Assessment of the Power Spectrum 1956.3.3 A Hierarchical Bayes Assessment of the Power Spectrum 1986.3.4 The Spectrum as a Covariate Using an Accelerated Life Model 2006.3.5 Closing Remarks on Signatures and Covariates 2027 Survival in Dynamic Environments 2057.1 Introduction: Why Stochastic Hazard Functions? 2057.2 Hazard Rate Processes 2067.2.1 Hazard Rates as Shot-noise Processes 2077.2.2 Hazard Rates as Lévy Processes 2087.2.3 Hazard Rates as Functions of Diffusion Processes 2107.3 Cumulative Hazard Processes 2117.3.1 The Cumulative Hazard as a Compound Poisson Process 2137.3.2 The Cumulative Hazard as an Increasing Lévy Process 2137.3.3 Cumulative Hazard as Geometric Brownian Motion 2147.3.4 The Cumulative Hazard as a Markov Additive Process 2157.4 Competing Risks and Competing Risk Processes 2187.4.1 Deterministic Competing Risks 2197.4.2 Stochastic Competing Risks and Competing Risk Processes 2207.5 Degradation and Aging Processes 2227.5.1 A Probabilistic Framework for Degradation Modeling 2237.5.2 Specifying Degradation Processes 2238 Point Processes for Event Histories 2278.1 Introduction: What is Event History? 2278.1.1 Parameterizing the Intensity Function 2298.2 Other Point Processes in Reliability and Life-testing 2298.2.1 Multiple Failure Modes and Competing Risks 2298.2.2 Items Experiencing Degradation and Deterioration 2318.2.3 Units Experiencing Maintenance and Repair 2318.2.4 Life-testing Under Censorship and Withdrawals 2338.3 Multiplicative Intensity and Multivariate Point Processes 2348.3.1 Multivariate Counting and Intensity Processes 2348.4 Dynamic Processes and Statistical Models: Martingales 2368.4.1 Decomposition of Continuous Time Processes 2388.4.2 Stochastic Integrals and a Martingale Central Limit Theorem 2398.5 Point Processes with Multiplicative Intensities 2409 Non-parametric Bayes Methods in Reliability 2439.1 The What and Why of Non-parametric Bayes 2439.2 The Dirichlet Distribution and its Variants 2449.2.1 The Ordered Dirichlet Distribution 2469.2.2 The Generalized Dirichlet – Concept of Neutrality 2469.3 A Non-parametric Bayes Approach to Bioassay 2479.3.1 A Prior for Potency 2489.3.2 The Posterior Potency 2499.4 Prior Distributions on the Hazard Function 2509.4.1 Independent Beta Priors on Piecewise Constant Hazards 2509.4.2 The Extended Gamma Process as a Prior 2519.5 Prior Distributions for the Cumulative Hazard Function 2539.5.1 Neutral to the Right Probabilities and Gamma Process Priors 2539.5.2 Beta Process Priors for the Cumulative Hazard 2559.6 Priors for the Cumulative Distribution Function 2599.6.1 The Dirichlet Process Prior 2609.6.2 Neutral to the Right-prior Processes 26410 Survivability of Co-operative, Competing and Vague Systems 26910.1 Introduction: Notion of Systems and their Components 26910.1.1 Overview of the Chapter 26910.2 Coherent Systems and their Qualitative Properties 27010.2.1 The Reliability of Coherent Systems 27410.3 The Survivability of Coherent Systems 28110.3.1 Performance Processes and their Driving Processes 28210.3.2 System Survivability Under Hierarchical Independence 28310.3.3 System Survivability Under Interdependence 28410.3.4 Prior Distributions on the Unit Hypercube 28610.4 Machine Learning Methods in Survivability Assessment 29110.4.1 An Overview of the Neural Net Methodology 29210.4.2 A Two-phased Neural Net for System Survivability 29310.5 Reliability Allocation: Optimal System Design 29410.5.1 The Decision Theoretic Formulation 29410.5.2 Reliability Apportionment for Series Systems 29610.5.3 Reliability Apportionment for Parallel Redundant Systems 29710.5.4 Apportioning Node Reliabilities in Networks 29810.5.5 Apportioning Reliability Under Interdependence 29810.6 The Utility of Reliability: Optimum System Selection 29910.6.1 Decision-making for System Selection 30010.6.2 The Utility of Reliability 30110.7 Multi-state and Vague Stochastic Systems 30310.7.1 Vagueness or Imprecision 30410.7.2 Many-valued Logic: A Synopsis 30510.7.3 Consistency Profiles and Probabilities of Vague Sets 30510.7.4 Reliability of Components in Vague Binary States 30710.7.5 Reliability of Systems in Vague Binary States 30710.7.6 Concluding Comments on Vague Stochastic Systems 30811 Reliability and Survival in Econometrics and Finance 30911.1 Introduction and Overview 30911.2 Relating Metrics of Reliability to those of Income Inequality 31011.2.1 Some Metrics of Reliability and Survival 31011.2.2 Metrics of Income Inequality 31111.2.3 Relating the Metrics 31311.2.4 The Entropy of Income Shares 31511.2.5 Lorenz Curve Analysis of Failure Data 31511.3 Invoking Reliability Theory in Financial Risk Assessment 31711.3.1 Asset Pricing of Risk-free Bonds: An Overview 31711.3.2 Re-interpreting the Exponentiation Formula 31911.3.3 A Characterization of Present Value Functions 32011.3.4 Present Value Functions Under Stochastic Interest Rates 32511.4 Inferential Issues in Asset Pricing 32811.4.1 Formulating the Inferential Problem 32911.4.2 A Strategy for Pooling Present Value Functions 32911.4.3 Illustrative Example: Pooling Present Value Functions 33111.5 Concluding Comments 332Appendix A Markov Chain Monté Carlo Simulation 335A.1 The Gibbs Sampling Algorithm 335Appendix B Fourier Series Models and the Power Spectrum 339B.1 Preliminaries: Trigonometric Functions 339B.2 Orthogonality of Trigonometric Functions 340B.3 The Fourier Representation of a Finite Sequence of Numbers 341B.4 Fourier Series Models for Time Series Data 342B.4.1 The Spectrum and the Periodgram of f(t) 343Appendix C Network Survivability and Borel’s Paradox 345C.1 Preamble 345C.2 Re-assessing Testimonies of Experts Who have Vanished 345C.3 The Paradox in Two Dimensions 346C.4 The Paradox in Network Survivability Assessment 347Bibliography 349Index 365