Empirical Laplace transform and approximation of compound distributions

Let {Xi}−r•â1â be i.i.d. non-negative random variables with d.f. F and Laplace transform L. Let N be integer valued and independent of {Xi}−r•â1â. In many applications one knows that for y⇒• and a function ℝrsquo;, P{Σ−rNâi=1âXi>y}∼ℝrsquo;(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 the authors derive an estimator τn for τ by solving ψ(τn,Ln(τn),L'