This chapter defines the COM–Poisson distribution in greater detail, discussing its associated attributes and computing tools available for analysis. This chapter first details how the COM–Poisson distribution was derived, and then describes the probability distribution, and introduces computing functions available in R that can be used to determine various probabilistic quantities of interest, including the normalizing constant, probability and cumulative distribution functions, random number generation, mean, and variance. The chapter then outlines the distributional and statistical properties associated with this model, and discusses parameter estimation and statistical inference associated with the COM–Poisson model. Various processes for generating random data are then discussed, along with associated available R computing tools. Continued discussion provides reparametrizations of the density function that serve as alternative forms for statistical analyses and model development, considers the COM–Poisson as a weighted Poisson distribution, and details discussion describing the various ways to approximate the COM–Poisson normalizing function.

This chapter provides an overview of operational risk modeling techniques used by industry participants and regulators in the USA, recommendations for how modeling techniques can be improved, and a summary of the model risk tools necessary for any operational risk modeling framework.

Stress-testing models pose a unique set of challenges with respect to performance monitoring. In particular, unlike standard forecasting models that generate unconditional forecasts, stress-testing models generate conditional forecasts based on stress scenarios that are unlikely to occur. This critical difference greatly limits one’s ability to assess model projections with observed outcomes. We provide several different methods for this purpose

This chapter describes the current state of CCR management, modeling and validation as of the early 2020s. Beginning with the historical evolution of counter party credit risk measurement and management, it discusses backtesting and stress testing as applicable to counterparty credit risk.

This chapter focuses on the three types of testing that banks are supposed to conduct for their VaR models. These are conceptual soundness, outcomes analysis and benchmarking. This chapter reviews how these three aspects of validation can be applied to VaR models of banks’ trading activities. In the case of backtesting and benchmarking it demonstrates how banks’ VaR models fare under some the backtesting and benchmarking tests.

This chapter is an overview summarizing relevant established and well-studied distributions for count data that motivate consideration of the Conway–Maxwell–Poisson distribution. Each of the discussed models provides an improved flexibility and computational ability for analyzing count data, yet associated restrictions help readers to appreciate the need for and usefulness of the Conway–Maxwell–Poisson distribution, thus resulting in an explosion of research relating to this model. For completeness of discussion, each of these sections includes discussion of the relevant R packages and their contained functionality to serve as a starting point for forthcoming discussions throughout subsequent chapters. Along with the R discussion, illustrative examples aid readers in understanding distribution qualities and related statistical computational output. This background provides insights regarding the real implications of apparent data dispersion in count data models, and the need to properly address it.

This chapter presents several case studies in the validation of wholesale credit risk models. The steps for each case study include the following: (1) use of the model; (2) internal and external data; (3) model assumptions and methodologies; (4) model performance; (5) outcomes analysis; and (6) the quality and comprehensiveness of development documentation.