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

CHAPTERFOUR

Flexible Heavy-Tailed Severity Models: α-Stable Family

In this chapter, we present advanced discussion for risk modellers relating to flexible distributions. The intention of this chapter is to provide a clear understanding of the key features of a very flexible and widely applicable class of models in the operational risk (OpRisk) and insurance context, provided by the α-stable family.

In order to develop a basic understanding of this important family of severity models, we first present several fundamental properties of the primary building block of all these models, the α-stable family of models. The family of univariate α-stable severity models describes not only heavy tails and asymmetric behaviour but also the dependency on four parameters that allows for a wide range of skewness and kurtosis features in loss data to be modeled appropriately. In addition, as will be discussed later, the domain of attraction for such families of models will naturally correspond to considering the distributions from which limits of sums of i.i.d. random variables will converge weakly (in distribution) to laws which are non-degenerate.

To proceed, we first present a note on some of the notations utilized in this chapter:

• The α-stable severity distribution, denoted by S_α(x; β, γ, δ; TYPE), is parameterized by parameters (generically denoted) α, β, γ and δ. The different parameterizations of this family of distributions will be made explicit with a subscript index and also by the ...