Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk

CHAPTERTHREE

Heavy-Tailed Model Class Characterizations for LDA

In this chapter, several important concepts that will be utilized throughout the following chapters will be developed and presented. The chapter will start with an introduction to the class of asymptotic notations that are adopted widely in the actuarial and mathematical literature. Then a high level presentation of the key components of characterizing tails of distributions will be presented. In particular, we will consider the many varied ways that one may classify a distribution function as being heavy tailed, this will allow a solid mathematical footing to be placed on the discussion about LDA models with heavy-tailed severity model and the effect this may have on modeling and capital estimation. We do not aim to provide a comprehensive coverage of this topic as there are already many great books on this topic, see, for instance, Bingham et al. (1989) and the more recent book by Borovkov and Borovkov (2002) to name a few. Instead we will provide in this chapter a guided tour through some important key results that will be applicable to both practitioners and researchers alike when investigating key aspects of capital estimation for operational risk (OpRisk) settings through asymptotic analysis. The following key topics are discussed:

1. Asymptotic notations;
2. Heavy-tailed models in the sub-exponential distribution class;
3. Heavy-tailed models in the regular and slow variation distribution classes;
4. Refinements of heavy-tailed ...