Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T22:35:44.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix-Form Recursions for a Family of Compound Distributions

Published online by Cambridge University Press:  09 August 2013

Xueyuan Wu  and
Shuanming Li
Xueyuan Wu
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Victoria 3010, Australia, E-mail: [email protected], Fax: 61-3-83446899
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Victoria 3010, Australia, E-mail: [email protected], Fax: 61-3-83446899
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

Keywords

Discrete phase-type distributionsGeneralised (a, b, 0) familyRecursive formulaCompound distribution
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 40 , Issue 1 , May 2010 , pp. 351 - 368
Copyright
Copyright © International Actuarial Association 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1992) Phase-type representations in random walk and queuing problems. Annals of Probability 20, 772–789.CrossRefGoogle Scholar
Asmussen, S. (2000) Ruin probabilities. World Scientific, Singapore.Google Scholar
Bobbio, A., Horváth, A., Scarpa, M. and Telek, M. (2003) Acyclic discrete phase type distributions: properties and a parameter estimation algorithm. Performance Evaluation 54, 1–32.CrossRefGoogle Scholar
Dickson, D.C.M. (2005) Insurance Risk and Ruin. International Series on Actuarial Science, Cambridge University Press.Google Scholar
Drekic, S., Dickson, D.C.M., Stanford, D.A. and Willmot, G.E. (2004) On the distribution of the deficit at ruin when claims are phase-type. Scandinavian Actuarial Journal, 3, 105–120.CrossRefGoogle Scholar
Eisele, K.T. (2006) Recursions for compound phase distributions. Insurance: Mathematics and Economics 38, 149–156.Google Scholar
Hesselager, O. (1996) Recursions for certain bivariate counting distributions and their compound distributions. ASTIN Bulletin 32, 283–297.Google Scholar
Hipp, C. (2006) Speedy convolution algorithms and Panjer recursions for phase-type distributions. Insurance: Mathematics and Economics 38, 176–188.Google Scholar
Latouche, G. and Ramaswami, V. (1999) Introduction to matrix analytic methods in stochastic modeling. ASA SIAM, Philadelphia.CrossRefGoogle Scholar
Neuts, M.F. (1975) Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin. University of Louvain, pp. 173–206.Google Scholar
Neuts, M.F. (1981) Matrix-geometric solutions in stochastic models: An algorithmic approach. Johns Hopkins University Press, Baltimore.Google Scholar
Ng, A.C.Y. and Yang, H. (2005) Lundberg-type bounds for the joint distribution of surplus immediate before and at ruin under the Sparre Andersen Model. North American Actuarial Journal 9(2), 85–107.Google Scholar
Panjer, H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22–26.Google Scholar
Schröter, K.J. (1991) On a family of counting distributions and recursions for the related compound distributions. Scandinavian Actuarial Journal, 161–175.Google Scholar
Stanford, D.A. and Stroiński, K.J. (1994) Recursive methods for computing finite-time ruin probabilities for phase-distributed claim sizes. ASTIN Bulletin 24, 235–254.CrossRefGoogle Scholar
Sundt, B. (1992) On some extensions of Panjer's class of counting distributions. ASTIN Bulletin 22, 61–80.CrossRefGoogle Scholar
Sundt, B. (1999) On multivariate Panjer recursions. ASTIN Bulletin 29, 29–45.Google Scholar
Sundt, B. (2002) Recursive evaluation of aggregate claims distributions. Insurance: Mathematics and Economics 30, 297–332.Google Scholar
Sundt, B. (2003) Some recursions for moments of compound distributions. Insurance: Mathematics and Economics 33, 486–496.Google Scholar
Sundt, B. and Jewell, W.S. (1981) Further results on recursive evaluation of compound distributions. ASTIN Bulletin 12, 27–39.Google Scholar