Preface. Part 1: Introduction. 1. Why do a risk analysis? 1.1. Moving on from "What If" Scenarios. 1.2. The Risk Analysis Process. 1.3. Risk Management Options. 1.4. Evaluating Risk Management Options. 1.5. Inefficiencies in Transferring Risks to Others. 1.6. Risk Registers. 2. Planning a risk analysis. 2.1. Questions and Motives. 2.2. Determine the Assumptions that are Acceptable or Required. 2.3. Time and Timing. 2.4. You'll Need a Good Risk Analyst or Team. 3. The quality of a risk analysis. 3.1. The Reasons Why a Risk Analysis can be Terrible. 3.2. Communicating the Quality of Data Used in a Risk Analysis. 3.3. Level of Criticality. 3.4. The Biggest Uncertainty in a Risk Analysis. 3.5. Iterate. 4. Choice of model structure. 4.1. Software Tools and the Models they Build. 4.2. Calculation Methods. 4.3. Uncertainty and Variability. 4.4. How Monte Carlo Simulation Works. 4.5. Simulation Modelling. 5. Understanding and using the results of a risk analysis. 5.1. Writing a Risk Analysis Report. 5.2. Explaining a Model's Assumptions. 5.3. Graphical Presentation of a Model's Results. 5.4. Statistical Methods of Analysing Results. Part 2: Introduction. 6. Probability mathematics and simulation. 6.1. Probability Distribution Equations. 6.2. The Definition of "Probability". 6.3. Probability Rules. 6.4. Statistical Measures. 7. Building and running a model. 7.1. Model Design and Scope. 7.2. Building Models that are Easy to Check and Modify. 7.3. Building Models that are Efficient. 7.4. Most Common Modelling Errors. 8. Some basic random processes. 8.1. Introduction. 8.2. The Binomial Process. 8.3. The Poisson Process. 8.4. The Hypergeometric Process. 8.5. Central Limit Theorem. 8.6. Renewal Processes. 8.7. Mixture Distributions. 8.8. Martingales. 8.9. Miscellaneous Example. 9. Data and statistics. 9.1. Classical Statistics. 9.2. Bayesian Inference. 9.3. The Bootstrap. 9.4. Maximum Entropy Principle. 9.5. Which Technique Should You Use? 9.6. Adding uncertainty in Simple Linear Least-Squares Regression Analysis. 10. Fitting distributions to data. 10.1. Analysing the Properties of the Observed Data. 10.2. Fitting a Non-Parametric Distribution to the Observed Data. 10.3. Fitting a First-Order Parametric Distribution to Observed Data. 10.4. Fitting a Second-Order Parametric Distribution to Observed Data. 11. Sums of random variables. 11.1. The Basic Problem. 11.2. Aggregate Distributions. 12. Forecasting with uncertainty. 12.1. The Properties of a Time Series Forecast. 12.2. Common Financial Time Series Models. 12.3. Autoregressive Models. 12.4. Markov Chain Models. 12.5. Birth and Death Models. 12.6. Time Series Projection of Events Occurring Randomly in Time. 12.7. Time Series Models with Leading Indicators. 12.8. Comparing Forecasting Fits for Different Models. 12.9. Long-Term Forecasting. 13. Modelling correlation and dependencies. 13.1. Introduction. 13.2. Rank Order Correlation. 13.3. Copulas. 13.4. The Envelope Method. 13.5. Multiple Correlation Using a Look-Up Table. 14. Eliciting from expert opinion. 14.1. Introduction. 14.2. Sources of Error in Subjective Estimation. 14.3. Modelling Techniques. 14.4. Calibrating Subject Matter Experts. 14.5. Conducting a Brainstorming Session. 14.6. Conducting the Interview. 15. Testing and modelling causal relationships. 15.1. Campylobacter Example. 15.2. Types of Model to Analyse Data. 15.3. From Risk Factors to Causes. 15.4. Evaluating Evidence. 15.5. The Limits of Causal Arguments. 15.6. An Example of a Qualitative Causal Analysis. 15.7. Is Causal Analysis Essential? 16. Optimisation in risk analysis. 16.1. Introduction. 16.2. Optimisation Methods. 16.3. Risk Analysis Modelling and Optimisation. 16.4. Working Example: Optimal Allocation of Mineral Pots. 17. Checking and validating a model. 17.1. Spreadsheet Model Errors. 17.2. Checking Model Behaviour. 17.3. Comparing Predictions Against Reality. 18. Discounted cashflow modelling. 18.1. Useful Time Series Models of Sales and Market Size. 18.2. Summing Random Variables. 18.3. Summing Variable Margins on Variable Revenues. 18.4. Financial Measures in Risk Analysis. 19. Project risk analysis. 19.1. Cost Risk Analysis. 19.2. Schedule Risk Analysis. 19.3. Portfolios of risks. 19.4. Cascading Risks. 20. Insurance and finance risk analysis modelling. 20.1. Operational Risk Modelling. 20.2. Credit Risk. 20.3. Credit Ratings and Markov Chain Models. 20.4. Other Areas of Financial Risk. 20.5. Measures of Risk. 20.6. Term Life Insurance. 20.7. Accident Insurance. 20.8. Modelling a Correlated Insurance Portfolio. 20.9. Modelling Extremes. 20.10. Premium Calculations. 21. Microbial food safety risk assessment. 21.1. Growth and Attenuation Models. 21.2. Dose-Response Models. 21.3. Is Monte Carlo Simulation the Right Approach? 21.4. Some Model Simplifications. 22. Animal import risk assessment. 22.1. Testing for an Infected Animal. 22.2. Estimating True Prevalence in a Population. 22.3. Importing Problems. 22.4. Confidence of Detecting an Infected Group. 22.5. Miscellaneous Animal Health and Food Safety Problems. I. Guide for lecturers. II. About ModelRisk. III. A compendium of distributions. III.1. Discrete and Continuous Distributions. III.2. Bounded and Unbounded Distributions. III.3. Parametric and Non-Parametric Distributions. III.4. Univariate and Multivariate Distributions. III.5. Lists of Applications and the Most Useful Distributions. III.6. How to Read Probability Distribution Equations. III.7. The Distributions. III.8. Introduction to Creating Your Own Distributions. III.9. Approximation of One Distribution with Another. III.10. Recursive Formulae for Discrete Distributions. III.11. A Visual Observation On The Behaviour Of Distributions. IV. Further reading. V. Vose Consulting. References. Index.

Risk Analysis: A Quantitative Guide

A survey of the application of gamma processes in maintenance

Engineers in the manufacturing industries have used accelerated test (AT) experiments for many decades. The purpose of AT experiments is to acquire reliability information quickly. Test units of a material, component, subsystem, or entire systems are subjected to higher-than-usual levels of one or more accelerating variables such as temperature or stress. Then the AT results are used to predict life of the units at use conditions. The extrapolation is typically justified (correctly or incorrectly) on the basis of physically motivated models or a combination of empirical model fitting with a sufficient amount of previous experience in testing similar units. The need to extrapolate in both time and the accelerating variables generally necessitates the use of fully parametric models. Statisticians have made important contributions in the development of appropriate stochastic models for AT data (typically a distribution for the response and regression relationships between the parameters of this distribution and the accelerating variable(s)), statistical methods for AT planning (choice of accelerating variable levels and allocation of available test units to those levels), and methods of estimation of suitable reliability metrics. This paper provides a review of many of the AT models that have been use successfully in this area.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1027&context=stat_las_preprints

A Review of Accelerated Test Models

The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.

https://www.jds-online.com/files/JDS-1210.pdf

The Weibull-G Family of Probability Distributions

: The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More importantly the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

Estimation for a family of life distributions with applications to fatigue

: A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

A new family of life distributions

Summary The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

Sam C. Saunders

Papers

A new family of life distributions

Estimation for a family of life distributions with applications to fatigue

Estimation for a family of life distributions