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Analysis of a Multivariate Claim Process

Author

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  • Qi-Ming He

    (University of Waterloo)

  • Jiandong Ren

    (University of Western Ontario)

Abstract

The first part of this paper introduces a class of discrete multivariate phase-type (MPH) distributions. Recursive formulas are found for joint probabilities. Explicit expressions are obtained for means, variances and co-variances. The discrete MPH-distributions are used in the second part of the paper to study multivariate insurance claim processes in risk analysis, where claims may arrive in batches, the arrivals of different types of batches may be correlated and the amounts of different types of claims in a batch may be dependent. Under certain conditions, it is shown that the total amounts of claims accumulated in some random time horizon are discrete MPH random vectors. Matrix-representations of the discrete MPH-distributions are constructed explicitly. Efficient computational methods are developed for computing risk measures of the total claims of different types of claim batches and individual types of claims (e.g., joint distribution, mean, variance, correlation and value at risk.)

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9420-9
    DOI: 10.1007/s11009-014-9420-9
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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.
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    4. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    5. 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.
    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.
    7. V. Ramaswami & Douglas Woolford & David Stanford, 2008. "The erlangization method for Markovian fluid flows," Annals of Operations Research, Springer, vol. 160(1), pages 215-225, April.
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    Cited by:

    1. Ren Jiandong & Zitikis Ricardas, 2017. "CMPH: a multivariate phase-type aggregate loss distribution," Dependence Modeling, De Gruyter, vol. 5(1), pages 304-315, December.
    2. Bladt, Martin & Yslas, Jorge, 2023. "Robust claim frequency modeling through phase-type mixture-of-experts regression," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 1-22.

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