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Some properties of convolutions of Pascal and Erlang random variables

Author

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  • Mi, J.
  • Shi, W.
  • Zhou, Y.Y.

Abstract

This article studies the convolutions of Pascal random variables, i.e., negative binomial distributions with integer parameters. We show that the probability distributions of such convolutions can be expressed as a generalized mixture (i.e., mixture with negative and positive coefficients) of a finite number of Pascal distribution functions. Based on this result, further study on the limiting behavior of the failure rate function of the convolution is presented. A sufficient condition is given to establish the likelihood ratio order between two convolutions of Pascal random variables. Similar results are obtained for the convolutions of Erlang random variables.

Suggested Citation

  • Mi, J. & Shi, W. & Zhou, Y.Y., 2008. "Some properties of convolutions of Pascal and Erlang random variables," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2378-2387, October.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:15:p:2378-2387
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    References listed on IDEAS

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    1. Boland, Philip J. & El-Neweihi, Emad & Proschan, Frank, 1994. "Schur properties of convolutions of exponential and geometric random variables," Journal of Multivariate Analysis, Elsevier, vol. 48(1), pages 157-167, January.
    2. Jie Mi, 1996. "Limiting behavior of mixtures of discrete lifetime distributions," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(3), pages 365-380, April.
    3. Furman, Edward, 2007. "On the convolution of the negative binomial random variables," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 169-172, January.
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    Cited by:

    1. Zhao, Peng, 2011. "Some new results on convolutions of heterogeneous gamma random variables," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 958-976, May.
    2. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    3. Zhao, Peng & Balakrishnan, N., 2010. "Ordering properties of convolutions of heterogeneous Erlang and Pascal random variables," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 969-974, June.
    4. Lu, ZhiYi & Meng, LiLi, 2011. "Stochastic comparisons for allocations of policy limits and deductibles with applications," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 338-343, May.

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