Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T01:05:30.988Z Has data issue: false hasContentIssue false
  • English
  • Français

EXPANSION IN BELL POLYNOMIALS OF THE DISTRIBUTION OF THE TOTAL CLAIM AMOUNT WITH WEIBULL-DISTRIBUTED CLAIM SIZES

Part of: Mathematical economics

Published online by Cambridge University Press:  01 April 2008

RAMON LACAYO
RAMON LACAYO*
Affiliation:
Escuela de Ingeniería Comercial, Universidad Católica del Norte, Larrondo 1281, Coquimbo, Chile (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.

Keywords

Laplace transformBell polynomialtotal claim amountclaim size processclaim number processprobability density function

MSC classification

Secondary: 91B30: Risk theory, insurance
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 4 , April 2008 , pp. 495 - 501
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Doetsch, G., Introduction to the theory and applications of the Laplace transformation (Springer, Berlin, 1974).CrossRefGoogle Scholar
[2]Grübel, R. and Hermesmeier, R., “Computation of compound distributions I: aliasing errors and exponential tilting”, Astin Bull. 29 (1999) 197–214.CrossRefGoogle Scholar
[3]Grübel, R. and Hermesmeier, R., “Computation of compound distributions II: discretization errors and Richardson extrapolation”, Astin Bull. 30 (2000) 309–331.CrossRefGoogle Scholar
[4]Encyclopaedia of mathematics (ed. M. Hazewinkel), (Kluwer, 1990).Google Scholar
[5]Mikosch, T., Non-life insurance mathematics. An introduction with stochastic processes (Springer, Berlin, 2004).Google Scholar
[6]Panjer, H. H., “Recursive evaluation of a family of compound distributions”, Astin Bull. 11 (1981) 22–26.CrossRefGoogle Scholar