Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T11:45:34.039Z Has data issue: false hasContentIssue false
  • English
  • Français

Inhomogeneous phase-type distributions and heavy tails

Part of: Markov processes Parametric inference Inference from stochastic processes

Published online by Cambridge University Press:  11 December 2019

Hansjörg Albrecher  and
Mogens Bladt
Hansjörg Albrecher*
Affiliation:
Université de Lausanne
Mogens Bladt*
Affiliation:
University of Copenhagen
*
*Postal address: Department of Actuarial Science, Université de Lausanne, Quartier UNIL-Chamberonne, Bâtiment Extranef, 1015 Lausanne, Switzerland.
**Postal address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.

Keywords

Inhomogeneous phase-typeheavy tailproduct integralmatrix Pareto distributionmatrix Weibull distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 62F10: Point estimation 62M05: Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1044 - 1064
Copyright
© Applied Probability Trust 2019 

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

Ahn, S., Kim, J. H. and Ramaswami, V. (2012). A new class of models for heavy tailed distributions in finance and insurance risk. Insurance Math. Econom. 51, 43–52.CrossRefGoogle Scholar
Albrecher, H., Beirlant, J. and Teugels, J. L. (2017). Reinsurance: Actuarial and Statistical Aspects. John Wiley, Chichester.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. Advanced Series on Statistical Science & Applied Probability, 14. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.CrossRefGoogle Scholar
Asmussen, S., Nerman, O. and Olsson, M. (1996). Fitting phase-type distributions via the EM algorithm. Scand. J. Statist. 23, 419–441.Google Scholar
Bladt, M. and Nielsen, B. F. (2017). Matrix-Exponential Distributions in Applied Probability. Springer, New York.CrossRefGoogle Scholar
Bladt, M., Nielsen, B. F. and Samorodnitsky, G. (2015). Calculation of ruin probabilities for a dense class of heavy tailed distributions. Scand. Actuarial J. 2015, 573–591.CrossRefGoogle Scholar
Bladt, M. and Rojas-Nandayapa, L. (2018). Fitting phase-type scale mixtures to heavy-tailed data and distributions. Extremes 21, 285–313.CrossRefGoogle Scholar
Breuer, L. (2012). From Markov Jump Processes to Spatial Queues. Springer Science & Business Media, New York.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18, 1501–1555.CrossRefGoogle Scholar
Goodman, G. S. and Johansen, S. (1973). Kolmogorov’s differential equations for non-stationary, countable-state Markov processes with uniformly continuous transition probabilities. Math. Proc. Camb. Phil. Soc. 73, 119–138.CrossRefGoogle Scholar
Helton, J. and Stuckwisch, S. (1976). Numerical approximation of product integrals. J. Math. Anal. Appl. 56, 410–437.CrossRefGoogle Scholar
Møller, C. M. M. (1992). Numerical evaluation of markov transition probabilities based on the discretized product integral. Scand. Actuarial J. 1992, 76–87.CrossRefGoogle Scholar
Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, New York.Google Scholar
Pigeon, M. and Denuit, M. (2011). Composite lognormal-Pareto model with random threshold. Scand. Actuarial J. 2011, 177–192.CrossRefGoogle Scholar
Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017). Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions. Insurance Math. Econom. 77, 65–77.CrossRefGoogle Scholar
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scand. Actuarial J. 2007, 20–33.CrossRefGoogle Scholar
Slavk, A. (2007). Product Integration. Its History and Applications. Matfyzpress, Prague.Google Scholar
Willmot, G. E. and Woo, J.-K. (2007). On the class of Erlang mixtures with risk theoretic applications. N. Amer. Actuarial J. 11, 99–115.CrossRefGoogle Scholar