Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

Chapter Seven Estimation of Frequency and Severity Models

Estimation of the frequency and severity distributions is a challenging task for low-frequency- /high-severity losses, due to very limited data for these risks. The main tasks involved in fitting the frequency and severity distributions using data are as follows:

• Finding the best point estimates for the distribution parameters;
• Quantification of the parameter uncertainties;
• Assessing the model quality (model error).

In general, these tasks can be accomplished by undertaking either a frequentist or a Bayesian approach. In this chapter, we present key aspects of each of these approaches. In addition, we note that such modeling paradigms can be performed in both parametric and non-parametric modeling frameworks, but here we focus primarily on a parametric modeling approach, typically adopted in OpRisk. In the context of parameteric modelling we cover components of estimation based on key statistical methods such as Maximum Likelihood Estimation (MLE), Expectation Maximization (EM) algorithm, Bayesian posterior inference methods such as Markov chain Monte Carlo (MCMC), Sequential Monte Carlo Samplers (SMC Samplers) as well as estimation in the presence of truncations. For a comprehensive overview of the nonparametric case, see a book-length review for Bayesian approaches Ghosh and Ramamoorthi (2003); Hjort et al. (2010), and for frequentist approaches, Van der Vaart (2000).

7.1 Frequentist Estimation

Fitting distribution ...