A wide class of heavy-tailed distributions and its applications

Let F(x) be a distribution function supported on [0, ∞) with an equilibrium distribution function F e (x). In this paper we pay special attention to the hazard rate function r e (x) of F e (x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of r e (x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., ℳ and ℳ*, where ℳ contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of ℳ and ℳ* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z − θ, where Z and θ are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class ℳ to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of r e (x).