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Multivariate Phase-Type Distributions

Author

Listed:
  • David Assaf

    (Hebrew University, Jerusalem, Israel)

  • Naftali A. Langberg

    (Haifa University, Haifa, Israel)

  • Thomas H. Savits

    (University of Pittsburgh, Pittsburgh, Pennsylvania)

  • Moshe Shaked

    (University of Arizona, Tucson, Arizona)

Abstract

A (univariate) random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

Suggested Citation

  • David Assaf & Naftali A. Langberg & Thomas H. Savits & Moshe Shaked, 1984. "Multivariate Phase-Type Distributions," Operations Research, INFORMS, vol. 32(3), pages 688-702, June.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:3:p:688-702
    DOI: 10.1287/opre.32.3.688
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    Citations

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    Cited by:

    1. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    2. Surya, Budhi Arta, 2022. "Conditional multivariate distributions of phase-type for a finite mixture of Markov jump processes given observations of sample path," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    3. Qi-Ming He & Jiandong Ren, 2016. "Analysis of a Multivariate Claim Process," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 257-273, March.
    4. Li, Haijun, 2003. "Association of multivariate phase-type distributions, with applications to shock models," Statistics & Probability Letters, Elsevier, vol. 64(4), pages 381-392, October.
    5. Weiwei Chen & Benjamin Melamed & Oleg Sokolinskiy & Ben Sopranzetti, 2017. "Cash Conversion Systems in Corporate Subsidiaries," Manufacturing & Service Operations Management, INFORMS, vol. 19(4), pages 604-619, October.
    6. Ren Jiandong & Zitikis Ricardas, 2017. "CMPH: a multivariate phase-type aggregate loss distribution," Dependence Modeling, De Gruyter, vol. 5(1), pages 304-315, December.
    7. Brigo, Damiano & Mai, Jan-Frederik & Scherer, Matthias, 2016. "Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 60-66.
    8. Bo Friis Nielsen, 2022. "Characterisation of multivariate phase type distributions," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 229-231, April.
    9. Qi-Ming He & Jiandong Ren, 2016. "Parameter Estimation of Discrete Multivariate Phase-Type Distributions," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 629-651, September.
    10. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    11. Cui, Lirong & Li, Haijun, 2007. "Analytical method for reliability and MTTF assessment of coherent systems with dependent components," Reliability Engineering and System Safety, Elsevier, vol. 92(3), pages 300-307.
    12. Woo, Jae-Kyung, 2016. "On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 354-363.
    13. Eisele, Karl-Theodor, 2008. "Recursions for multivariate compound phase variables," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 65-72, February.
    14. Asimit, Alexandru V. & Jones, Bruce L., 2007. "Extreme behavior of multivariate phase-type distributions," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 223-233, September.
    15. Haijun Li & Susan H. Xu, 2001. "Directionally Convex Comparison of Correlated First Passage Times," Methodology and Computing in Applied Probability, Springer, vol. 3(4), pages 365-378, December.
    16. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    17. Berdel, Jasmin & Hipp, Christian, 2011. "Convolutions of multivariate phase-type distributions," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 374-377, May.
    18. Cai, Jun & Li, Haijun, 2007. "Dependence properties and bounds for ruin probabilities in multivariate compound risk models," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 757-773, April.
    19. Roel Verbelen & Katrien Antonio & Gerda Claeskens, 2016. "Multivariate mixtures of Erlangs for density estimation under censoring," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 22(3), pages 429-455, July.

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