Empirical Laplace transform and approximation of compound distributions

TLDR: In this paper, an estimator τ n for τ was derived by solving ψ(τ n,L n (τ n ),L' n (φ n ),...)=0 where L n is the empirical version of L.

Abstract: Let (Xn) be a sequence of non-negative random variables with distribution function F and Laplace transform L, and let N be an integer independent of the sequence. In many applications one knows that for y→∞ and a function φ P{Σ i=1 N X i >y}∼φ(y,τ,L(τ),L'(τ),...) where in turn τ is the solution of an equation ψ(τ,L(τ),...)=0. On the basis of a sample of size n we derive an estimator τ n for τ by solving ψ(τ n ,L n (τ n ),L' n (τ n ),...)=0 where L n is the empirical version of L. This estimator is then used to derive the asymptotic behaviour of φ(y,τ n ,L n (τ n ),L' n (τ n ),...). We include examples from insurance mathematics