# Bayesian Reliability

## Data Sets and Exercise Solutions

Selected solutions for the book are compiled here.

Chapter 1: Reliability Concepts
This chapter introduces the fundamental definitions of reliability and gives examples of common types of reliability data.

Chapter 2: Bayesian Inference

In this chapter we review the fundamental concepts of Bayesian and likelihood-based inference in reliability. We explore prior distributions, sampling distributions, posterior distributions, and the relation between the three quantities as specified through Bayes' Theorem. We also provide examples of inference in both discrete and continuous settings.

Chapter 3: Advanced Bayesian Modeling and Computational Methods
We extend the model structures described in the previous chapters using Bayesian hierarchical models. Because we generally cannot write the posterior distributions that result from these more complicated models in closed form, we begin this chapter with a description of Markov chain Monte Carlo algorithms that can be used to generate samples from intractable posterior distributions. These samples provide the basis for subsequent model inference. We also discuss empirical Bayes' methods. Finally, we describe techniques for assessing the sensitivity of model inferences to prior assumptions and a broadly applicable model diagnostic.

Chapter 4: Component Reliability This chapter presents models for various types of component reliability data, which consist of sampling and prior distributions. Several examples with real data, including some for which the data are censored, illustrate the use of these models in assessing component reliability. The complexity of some of these examples requires the use of hierarchical models. This chapter also introduces methods for model selection.
Chapter 5: System Reliability This chapter extends the models for component data to systems. This extension requires us to specify logical relationships between the components in a system and how the functioning of the complete system depends on the functioning (or not) of each of its components. We consider models for both independent and dependent component failures.

Chapter 6: Repairable System Reliability This chapter considers the reliability of multiple-time-use systems that are repaired when they fail. The effectiveness of repairs varies from restoring a system to a brand new state to restoring it to the reliability just before the system last failed. Several models for failure count and failure time data collected on repairable systems allow for different degrees of repair effectiveness. The models considered include renewal processes, homogeneous and nonhomogeneous Poisson processes, modulated power law processes, and a piecewise exponential model. This chapter also addresses how well these models fit the data and evaluates current reliability and other performance criteria, which characterize the reliability of repairable systems.

Chapter 7: Regression Models in Reliability The distribution of reliability data may depend on covariates, also known as explanatory variables, independent variables, predictors, or regressors. This chapter shows how to incorporate covariates in the analysis of binomial success/failure data, Poisson count data, and lifetime data. Covariates allow us to compare the reliability between two or more different situations. We also discuss how covariates arise in accelerated life testing and in experiments to improve reliability.

Chapter 8: Using Degradation Data to Assess Reliability
While reliability analysts have long used lifetime data for product/process reliability assessments, they began to employ degradation data for the same purpose in the 1990s. Assessing reliability with degradation data has a number of advantages. The analyst does not have to wait for failures to occur and can use less acceleration to collect degradation data. This chapter explains how to assess reliability using degradation data and also discusses how to accommodate covariates such as acceleration factors that speed up degradation and experimental factors that impact reliability in reliability improvement experiments. We also consider situations in which degradation measurements are destructive and conclude by introducing alternative stochastic models for degradation data.
Chapter 9: Planning for Reliability Data Collection This chapter considers planning for reliability data collection. Data collection planning determines how to optimally collect data, given a limited amount of resources (typically, money, time, and the number of units to test). This chapter discusses various planning criteria and presents a simulation-based framework to evaluate these criteria. Depending on the situation, planning can involve single and multiple planning variables. For multiple planning variable situations, we show how to use a genetic algorithm to find a near-optimal plan. This chapter illustrates data collection planning for a number of problems involving binomial, lifetime, accelerated life test, degradation, and system reliability data.

Chapter 10: Assurance Testing Planning for Bayesian assurance testing involves determining a test plan that guarantees that a reliability-related quantity of interest meets or exceeds a specified requirement at a desired level of confidence. Within a Bayesian hierarchical framework, this chapter determines test plans for binomial, Poisson, and Weibull testing. Also, we develop Weibull assurance test plans using available data from an associated life testing program.
• Chapter 10 Data Sets