Reliability and Risk

"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 modelsworks as both an educational tool and as a reference." (MAA Reviews, March 6, 2007)

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

Preface xiii Acknowledgements xv 1 Introduction and Overview 1 1.1 Preamble: What do Reliability, Risk and Robustness Mean? 1 1.2 Objectives and Prospective Readership 3 1.3 Reliability, Risk and Survival: State-of-the-Art 3 1.4 Risk Management: A Motivation for Risk Analysis 4 1.5 Books on Reliability, Risk and Survival Analysis 6 1.6 Overview of the Book 7 2 The Quantification of Uncertainty 9 2.1 Uncertain Quantities and Uncertain Events: Their Definition and Codification 9 2.2 Probability: A Satisfactory Way to Quantify Uncertainty 10 2.2.1 The Rules of Probability 11 2.2.2 Justifying the Rules of Probability 12 2.3 Overview of the Different Interpretations of Probability 13 2.3.1 A Brief History of Probability 14 2.3.2 The Different Kinds of Probability 16 2.4 Extending the Rules of Probability: Law of Total Probability and Bayes Law 19 2.4.1 Marginalization 20 2.4.2 The Law of Total Probability 20 2.4.3 Bayes Law: The Incorporation of Evidence and the Likelihood 20 2.5 The Bayesian Paradigm: A Prescription for Reliability, Risk and Survival Analysis 22 2.6 Probability Models, Parameters, Inference and Prediction 23 2.6.1 The Genesis of Probability Models and Their Parameters 24 2.6.2 Statistical Inference and Probabilistic Prediction 26 2.7 Testing Hypotheses: Posterior Odds and Bayes Factors 27 2.7.1 Bayes Factors: Weight of Evidence and Change in Odds 28 2.7.2 Uses of the Bayes Factor 30 2.7.3 Alternatives to Bayes Factors 31 2.8 Utility as Probability and Maximization of Expected Utility 32 2.8.1 Utility as a Probability 32 2.8.2 Maximization of Expected Utility 33 2.8.3 Attitudes to Risk: The Utility of Money 33 2.9 Decision Trees and Influence Diagrams for Risk Analysis 34 2.9.1 The Decision Tree 34 2.9.2 The Influence Diagram 35 3 Exchangeability and Indifference 45 3.1 Introduction to Exchangeability: de Finettis Theorem 45 3.1.1 Motivation for the Judgment of Exchangeability 46 3.1.2 Relationship between Independence and Exchangeability 46 3.1.3 de Finettis Representation Theorem for Zero-one Exchangeable Sequences 48 3.1.4 Exchangeable Sequences and the Law of Large Numbers 49 3.2 de Finetti-style Theorems for Infinite Sequences of Non-binary Random Quantities 50 3.2.1 Sufficiency and Indifference in Zero-one Exchangeable Sequences 51 3.2.2 Invariance Conditions Leading to Mixtures of Other Distributions 51 3.3 Error Bounds on de Finetti-style Results for Finite Sequences of Random Quantities 55 3.3.1 Bounds for Finitely Extendable Zero-one Random Quantities 55 3.3.2 Bounds for Finitely Extendable Non-binary Random Quantities 56 4 Stochastic Models of Failure 59 4.1 Introduction 59 4.2 Preliminaries: Univariate, Multivariate and Multi-indexed Distribution Functions 59 4.3 The Predictive Failure Rate Function of a Univariate Probability Distribution 62 4.3.1 The Case of Discontinuity 65 4.4 Interpretation and Uses of the Failure Rate Function the Model Failure Rate 66 4.4.1 The True Failure Rate: Does it Exist? 69 4.4.2 Decreasing Failure Rates, Reliability Growth, Burn-in and the Bathtub Curve 69 4.4.3 The Retrospective (or Reversed) Failure Rate 74 4.5 Multivariate Analogues of the Failure Rate Function 76 4.5.1 The Hazard Gradient 76 4.5.2 The Multivariate Failure Rate Function 77 4.5.3 The Conditional Failure Rate Functions 78 4.6 The Hazard Potential of Items and Individuals 79 4.6.1 Hazard Potentials and Dependent Lifelengths 81 4.6.2 The Hazard Gradient and Conditional Hazard Potentials 83 4.7 Probability Models for Interdependent Lifelengths 85 4.7.1 Preliminaries: Bivariate Distributions 85 4.7.2 The Bivariate Exponential Distributions of Gumbel 89 4.7.3 Freunds Bivariate Exponential Distribution 91 4.7.4 The Bivariate Exponential of Marshall and Olkin 93 4.7.5 The Bivariate Pareto as a Failure Model 107 4.7.6 A Bivariate Exponent