Hostname: page-component-6766d58669-r8qmj Total loading time: 0 Render date: 2026-05-15T05:21:34.688Z Has data issue: false hasContentIssue false
  • English
  • Français

On the nature of Phase-type Poisson distributions

Published online by Cambridge University Press:  05 November 2013

Sophie Hautphenne ,
Guy Latouche  and
Giang T. Nguyen
Sophie Hautphenne
Affiliation:
The University of Melbourne, Department of Mathematics and Statistics, Victoria 3010, Australia
Guy Latouche
Affiliation:
Université libre de Bruxelles, Département d'Informatique, CP 212, Boulevard du Triomphe, 1050 Bruxelles, Belgium
Giang T. Nguyen*
Affiliation:
The University of Adelaide, School of Mathematical Sciences, SA 5005, Australia
*
*Correspondence to: Giang T. Nguyen, The University of Adelaide, School of Mathematical Sciences, SA 5005, Australia. E-mail: giang.nguyen@adelaide.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Matrix-form Poisson probability distributions were recently introduced as one matrix generalization of Panjer distributions. We show in this paper that under the constraint that their representation is to be nonnegative, they have a physical interpretation as extensions of PH distributions, and we name this restricted family Phase-type Poisson. We use our physical interpretation to construct an EM algorithm-based estimation procedure.

Keywords

91B30secondary 65Q3062P05.Panjer's algorithmgeneralized Panjer distributionscompound distributionsEM algorithmminimal variancePH-Poisson.

Information

Type
Papers
Information
Annals of Actuarial Science , Volume 8 , Issue 1 , March 2014 , pp. 79 - 98
Copyright
Copyright © Institute and Faculty of Actuaries 2013 

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.)

Article purchase

Temporarily unavailable

References

Ahlfors, L. (1979). Complex analysis: An introduction to the theory of analytic functions of one complex variable. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc.Google Scholar
Asmussen, S., Nerman, O., Olsson, M. (1996). Fitting Phase-Type distributions via the EM algorithm. Scandinavian Journal of Statistics, 23, 419–441.Google Scholar
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society: Series B, 39, 1–38.Google Scholar
Eisele, K.-T. (2006). Recursions for compound phase distributions. Insurance: Mathematics and Economics, 38, 149–156.Google Scholar
Knuth, D.E. (1968). The Art of Computer Programming: Fundamental Algorithms, vol. 1. Addison–Wesley, Reading, MA.Google Scholar
Latouche, G., Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia PA.Google Scholar
Latouche, G., Remiche, M., Taylor, P. (2003). Transient Markov arrival processes. Annals of Applied Probability, 13, 628–640.Google Scholar
Neuts, M.F. (1975). Probability distributions of Phase Type. In Liber Amicorum Prof. Emeritus H. Florin. University of Louvain, Belgium, 173–206.Google Scholar
Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
Nocedal, J., Wright, S.J. (2000). Numerical Optimization. Springer.Google Scholar
Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 12, 22–26.CrossRefGoogle Scholar
Ren, J. (2010). Recursive formulas for compound phase distributions – univariate and bivariate cases. ASTIN Bulletin, 40, 615–629.Google Scholar
Siaw, K.K., Wu, X., Pitt, D., Wang, Y. (2011). Matrix-form recursive evaluation of the aggregate claims distribution revisited. Annals of Actuarial Science, 5, 163–179.Google Scholar
Sundt, B., Jewell, W.S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12, 27–39.Google Scholar
Telek, M. (2000). The minimal coefficient of variation of discrete phase-type distributions. In Advances in Algorithmic Methods for Stochastic Models – Proceedings of the 3rd International Conference on Matrix-Analytic Methods, G. Latouche and P. Taylor, Eds. Notable Publications Inc, NJ, 391–400.Google Scholar
Wilf, H.S. (1993). The asymptotic behavior of the Stirling numbers of the first kind. Journal of Combinatorial Theory, Series A 64, 2, 344–349.CrossRefGoogle Scholar
Wu, C.F.J. (1983). On the convergence properties of the EM algorithm. Annals of Statistics, 11, 95–103.CrossRefGoogle Scholar
Wu, X., Li, S. (2010). Matrix-form recursions for a family of compound distributions. ASTIN Bulletin, 40, 351–368.Google Scholar