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Empirical Laplace Transform and Approximation of Compound Distributions

Sándor Csörgő and Jef L. Teugels
Journal of Applied Probability
Vol. 27, No. 1 (Mar., 1990), pp. 88-101
DOI: 10.2307/3214597
Stable URL: http://www.jstor.org/stable/3214597
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Empirical Laplace Transform and Approximation of Compound Distributions
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Abstract

Let {Xi}1 ∞ 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}1 ∞. In many applications one knows that for y → ∞ and a function φ $P\left\{ \sum_{i=1}^{N}\,X_{i}>y\right\} \sim \varphi (y,\tau ,L(\tau),L^{\prime }(\tau ),\cdots )$ 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 τn for τ by solving ψ(τn, Ln(τn), Ln ′(τn),⋯) = 0 where Ln is the empirical version of L. This estimator is then used to derive the asymptotic behaviour of φ(y, τn, Ln(τn), Ln ′(τn),⋯). We include five examples, some of which are taken from insurance mathematics.

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