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Density and distribution evaluation for convolution of independent gamma variables

Author

Listed:
  • Chaoran Hu

    (University of Connecticut)

  • Vladimir Pozdnyakov

    (University of Connecticut)

  • Jun Yan

    (University of Connecticut)

Abstract

Convolutions of independent gamma variables are encountered in many applications such as insurance, reliability, and network engineering. Accurate and fast evaluations of their density and distribution functions are critical for such applications, but no open source, user-friendly software implementation has been available. We review several numerical evaluations of the density and distribution of convolution of independent gamma variables and compare them with respect to their accuracy and speed. The methods that are based on the Kummer confluent hypergeometric function are computationally most efficient. The benefit of employing the confluent hypergeometric function is further demonstrated by a renewal process application, where the gamma variables in the convolution are Erlang variables. We derive a new computationally efficient formula for the probability mass function of the number of renewals by a given time. An R package coga provides efficient C++ based implementations of the discussed methods and are available in CRAN.

Suggested Citation

  • Chaoran Hu & Vladimir Pozdnyakov & Jun Yan, 2020. "Density and distribution evaluation for convolution of independent gamma variables," Computational Statistics, Springer, vol. 35(1), pages 327-342, March.
    • Handle: RePEc:spr:compst:v:35:y:2020:i:1:d:10.1007_s00180-019-00924-9
    DOI: 10.1007/s00180-019-00924-9
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    References listed on IDEAS

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    1. Sim, C. H., 1992. "Point processes with correlated gamma interarrival times," Statistics & Probability Letters, Elsevier, vol. 15(2), pages 135-141, September.
    2. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, October.
    3. Sen, Ananda & Balakrishnan, N., 1999. "Convolution of geometrics and a reliability problem," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 421-426, July.
    4. Furman, Edward & Landsman, Zinoviy, 2005. "Risk capital decomposition for a multivariate dependent gamma portfolio," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 635-649, December.
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    Cited by:

    1. Paulusch, Joachim & Schlütter, Sebastian, 2022. "Sensitivity-implied tail-correlation matrices," Journal of Banking & Finance, Elsevier, vol. 134(C).

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