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Association of multivariate phase-type distributions, with applications to shock models

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  • Li, Haijun

Abstract

A random vector is said to be of (multivariate) phase-type if it can be represented as the vector of random times until absorptions into various stochastically closed subsets of the finite state space in an absorbing Markov chain. The phase-type distributions are useful since Markovian methods may be applicable in the situations where one adopts a (univariate or multivariate) phase-type distribution for time intervals that are needed in setting up a stochastic model. This paper studies the dependence nature of multivariate phase-type distributions and some related shock models, and it shows that under some mild conditions, the multivariate phase-type distributions are positively associated. The association properties for the lifetimes of components operating in some common shock environments are also obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:64:y:2003:i:4:p:381-392
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    References listed on IDEAS

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    1. V. G. Kulkarni, 1989. "A New Class of Multivariate Phase Type Distributions," Operations Research, INFORMS, vol. 37(1), pages 151-158, February.
    2. Li, Haijun & Xu, Susan H., 2001. "Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model," Journal of Multivariate Analysis, Elsevier, vol. 76(1), pages 63-89, January.
    3. Moshe Shaked, 1984. "Extensions of the Freund Distribution with Applications in Reliability Theory," Operations Research, INFORMS, vol. 32(4), pages 917-925, August.
    4. Shaked, Moshe & George Shanthikumar, J., 1987. "The multivariate hazard construction," Stochastic Processes and their Applications, Elsevier, vol. 24(2), pages 241-258, May.
    5. Lindqvist, Bo Henry, 1988. "Association of probability measures on partially ordered spaces," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 111-132, August.
    6. 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.
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    Cited by:

    1. 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.
    2. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    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. Haijun Li & Susan Xu & Way Kuo, 2014. "Asymptotic analysis of simultaneous damages in spatial Boolean models," Annals of Operations Research, Springer, vol. 212(1), pages 139-154, January.

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