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

9.6.2 SUBFAMILIES OF THE GENERALIZED HYPERBOLIC SEVERITY MODELS

As noted by Barndorff-Nielsen and Stelzer (2005), the GH family of models contains several well-known subclasses of parametric severity models given by different values of v such as in the following cases:

1. v = 1 one obtains the subfamily of hyperbolic distributions (see Example 9.6);
2. v = one obtains the subfamily of NIG distributions;
3. Other distributional subfamilies include Gaussian, Exponential, Laplace, Variance-Gamma and Student-t.

EXAMPLE 9.6 Examples of Flexible Hyperbolic Distributions (GH with v = 1)

In the Figure 9.8 plots and we explore the density shapes for the case of the sub-family of GH distributions given by the hyperbolic distributions where v = 1.

Figure 9.8 Top subplot: parameters used in this subplot study the effect of the shape parameter ranges α = [0.5, 1, 2, 5], skewness parameter β = 0, location parameters μ = 5, and scale parameter δ = 10. Bottom subplot: parameters used in this subplot study the effect of the skewness parameter ranges β = [0.1, 0.25, 0.5, 0.75], shape parameter α = 1, location parameters μ = 5, and scale parameter δ = 10

One of the representations of the GH family ...