Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T20:25:41.945Z Has data issue: false hasContentIssue false
  • English
  • Français

A Review on Phase-type Distributions and their Use in Risk Theory

Published online by Cambridge University Press:  17 April 2015

Mogens Bladt
Mogens Bladt*
Affiliation:
IIMAS-UNAM, A.P. 20-726, 01000 Mexico, D.F. – Mexico, E-mail: bladt@sigma.iimas.unam.mx
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.

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

Keywords

Phase-type distributionruin probabilityEM-algorithmMarkov chain Monte Carlo
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 145 - 161
Copyright
Copyright © ASTIN Bulletin 2005

References

Aalen, O.O. (1995) Phase type distributions in survival analysis. Scand. J. Statist., 22, 447–463.Google Scholar
Asmussen, S. (2000) Ruin Probabilities, volume 2 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Inc., River Edge, N.J. CrossRefGoogle Scholar
Asmussen, S. (2003) Applied Probability and Queues. Springer-Verlag, New York.Google Scholar
Asmussen, S. and Bladt, M. (1996) Phase-type distribution and risk processes with state-dependent premiums. Scand. Actuar. J., 1, 19–36.CrossRefGoogle Scholar
Asmussen, S. and Bladt, M. (1997) Renewal theory and queueing algorithms for matrix-exponential distributions. In Chakravarthy, S. and Alfa, A., editors, Matrix-analytic methods in stochastic models, volume 183 of Lecture Notes in Pure and Appl. Math., pages 313–341. Dekker, New York.Google 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
Asmussen, S., Avram, F. and Usabel, M. (2002) Erlangian approximations for finite-horizon ruin probabilities. Astin Bulletin, 32(2), 267–281.CrossRefGoogle Scholar
Bladt, M. and Neuts, M.F. (2003) Matrix-exponential distributions: calculus and interpretations via flows. Stochastic Models, 19(1), 113–124.CrossRefGoogle Scholar
Bladt, M., Gonzalez, A. and Lauritzen, S.L. (2002) The estimation of phase-type related functionals through Markov chain Monte Carlo methods. Scand. Actuar. J., 4, 280–300.Google Scholar
Erlang, A. (1909) Sandsynlighedsregning og telefonsamtaler. Nyt tidsskrift for Matematik, 20, 33–39.Google Scholar
Jensen, A. (1953) A distribution model applicable to economics. Munksgaard, Copenhagen.Google Scholar
Latouche, G. and Ramaswami, V. (1999) Introduction to matrix analytic methods in stochastic modeling. ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.Google Scholar
Moler, C. and Van Loan, C. (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev., 20, 801–836.CrossRefGoogle Scholar
Neuts, M.F. (1981) Matrix-geometric solutions in stochastic models., volume 2 of John Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, Md.Google Scholar
Neuts, M.F. (1989) Structured stochastic matrices of the M/G/1 type and their applications, volume 5 of Probability: Pure and Applied. Marcel Dekker Inc., New York.Google Scholar
Neuts, M.F. (1995) Algorithmic probability. Stochaistic Modeling Series. Chapman & Hall, London.Google Scholar
Oakes, D. (1999) Direct calculation of the information matrix via the EM algorithm. J. Royal Statist. Soc. B, 61, 479–482.CrossRefGoogle Scholar
O’Cinneide, C.A. (1990) Characterization of phase-type distributions. Comm. Statist. Stochastic Models, 6, 1–57.CrossRefGoogle Scholar
Press, W., Teukolsky, S., Vetterling, W. and Flannery, B. (1992) Numerical Recipes in Fortran. Cambridge University Press.Google Scholar