Approximation and estimation of some compound distributions |
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Authors: | Jozef L. Teugels |
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Affiliation: | Katholieke Universiteit Leuven, Leuven, Belgium |
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Abstract: | The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution Gt(x) = Σpn(t)F(n)(x) where pn(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of Gt(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - Gt(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate. |
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Keywords: | Compound distributions Poisson process Subexponential distributions Total claim amount |
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