IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v32y1984i3p688-702.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.32.3.688
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.32.3.688?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Eisele, Karl-Theodor, 2008. "Recursions for multivariate compound phase variables," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 65-72, February.
    6. 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.
    7. 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.
    8. 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.
    9. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    10. 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).
    11. 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.
    12. 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.
    13. Ren Jiandong & Zitikis Ricardas, 2017. "CMPH: a multivariate phase-type aggregate loss distribution," Dependence Modeling, De Gruyter, vol. 5(1), pages 304-315, December.
    14. 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.
    15. Bo Friis Nielsen, 2022. "Characterisation of multivariate phase type distributions," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 229-231, April.
    16. 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.
    17. 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.
    18. Berdel, Jasmin & Hipp, Christian, 2011. "Convolutions of multivariate phase-type distributions," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 374-377, May.
    19. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:32:y:1984:i:3:p:688-702. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.