Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 629–651 | Cite as

Parameter Estimation of Discrete Multivariate Phase-Type Distributions

  • Qi-Ming HeEmail author
  • Jiandong Ren


This paper considers parameter estimation of a class of discrete multi-variate phase-type distributions (DMPH). Such discrete phase-type distributions are based on discrete Markov chains with marked transitions introduced by He and Neuts (Stoch Process Appl 74(1):37–52, 1998) and is a generalization of the discrete univariate phase–type distributions. Properties of the DMPHs are presented. An EM-algorithm is developed for estimating the parameters for DMPHs. A number of numerical examples are provided to address some interesting parameter selection issues and to show possible applications of DMPHs.


Risk analysis Parameter estimation PH-distributions 

Mathematics Subject Classification (2010)

Primary: 60-08 Secondary: 60J22 


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  1. Asmussen S, Nerman O, Olsson M (1996) Fitting phase-type distributions via the EM-algorithm. Scand J Stat 23(4):419–441zbMATHGoogle Scholar
  2. Assaf D, Langberg NA, Savits TH, Shaked M (1984) Multivariate phase-type distributions. Oper Res 32(3):688–702MathSciNetCrossRefzbMATHGoogle Scholar
  3. Basawa IV, Rao PBLS (1980) Statistical inference for stochastic processes. Academic Press, London ; TorontozbMATHGoogle Scholar
  4. Breuer L (2003) An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Ann Oper Res 112:123–138MathSciNetCrossRefzbMATHGoogle Scholar
  5. Buchholz P, Kemper P, Kriege J (2010) Multi-class Markovian arrival processes and their parameter fitting. Perform Eval 67(11):1092–1106CrossRefGoogle Scholar
  6. Cai J, Li H (2005) Conditional tail expectations for multivariate phase-type distributions. J Appl Probab 42(3):810–825MathSciNetCrossRefzbMATHGoogle Scholar
  7. David Cummins J, Wiltbank LJ (1983) Estimating the total claims distribution using multivariate frequency and severity distributions. The Journal of Risk and Insurance 50(3):377–403CrossRefGoogle Scholar
  8. He Q-M, Neuts MF (1998) Markov chains with marked transitions. Stoch Process Appl 74(1):37–52MathSciNetCrossRefzbMATHGoogle Scholar
  9. He Q-M, Ren J (2014) Analysis of a multivariate claim process. Methodol Comput Appl Probab. (Accepted)Google Scholar
  10. Herbertsson A (2011) Modelling default contagion using multivariate phase-type distributions. Rev Deriv Res 14(1):1–36CrossRefzbMATHGoogle Scholar
  11. Kulkarni VG (1989) A new class of multivariate phase type distributions. Oper Res 37(1):151–158MathSciNetCrossRefzbMATHGoogle Scholar
  12. Olsson M (1996) Estimation of phase-type distributions from censored data. Scand J Stat 23:443–460MathSciNetzbMATHGoogle Scholar
  13. Ren J (2010) Recursive formulas for compound phase distributions. ASTIN Bulletin-Actuarial Studies in non LifeInsurance 40(2):615zbMATHGoogle Scholar
  14. Jeff Wu CF (1983) On the convergence properties of the EM algorithm. Ann Stat 11(1):95–103MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hassan Zadeh A., Bilodeau M (2013) Fitting bivariate losses with phase type distributions. Scand Actuar J 2013(4):241–262CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of WaterlooWaterlooCanada
  2. 2.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada

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