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Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables

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  • Zhao, Peng
  • Balakrishnan, N.

Abstract

In this paper, we study the convolutions of heterogeneous exponential and geometric random variables in terms of the weakly majorization order () of parameter vectors and the likelihood ratio order (>=lr). It is proved that order between two parameter vectors implies >=lr order between convolutions of two heterogeneous exponential (geometric) samples. For the two-dimensional case, it is found that there exist stronger equivalent characterizations. These results strengthen the corresponding ones of Boland et al. [Boland, P.J., El-Neweihi, E., Proschan, F., 1994. Schur properties of convolutions of exponential and geometric random variables. Journal of Multivariate Analysis 48, 157-167] by relaxing the conditions on parameter vectors from the majorization order () to order.

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  • Zhao, Peng & Balakrishnan, N., 2009. "Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1717-1723, August.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:15:p:1717-1723
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    References listed on IDEAS

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    1. Sen, Ananda & Balakrishnan, N., 1999. "Convolution of geometrics and a reliability problem," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 421-426, July.
    2. Korwar, Ramesh M., 2002. "On Stochastic Orders for Sums of Independent Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 344-357, February.
    3. Kochar, Subhash & Ma, Chunsheng, 1999. "Dispersive ordering of convolutions of exponential random variables," Statistics & Probability Letters, Elsevier, vol. 43(3), pages 321-324, July.
    4. 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.
    5. Zhao, Peng & Li, Xiaohu & Balakrishnan, N., 2009. "Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 952-962, May.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.

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