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Monotonicity and aging properties of random sums

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  • Cai, Jun
  • Willmot, Gordon E.

Abstract

In this paper, we discuss the distributional properties of random sums. We first derive conditions under which the distribution of a binomial sum is PF2 and then show under the same conditions the distribution of a Poisson sum is PF2 by approximating a Poisson sum by a sequence of binomial sums. The PF2 property reveals the monotonicity property of the reversed failure rates of certain compound Poisson distributions. Further, we discuss a class of random sums and derive the NWUE aging property of random sums in the class. The result, together with existing results, shows that the aging properties of many random sums can be characterized uniquely by the aging properties of the primary distributions of the random sums whatever the underlying distributions of the random sums are.

Suggested Citation

  • Cai, Jun & Willmot, Gordon E., 2005. "Monotonicity and aging properties of random sums," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 381-392, July.
    • Handle: RePEc:eee:stapro:v:73:y:2005:i:4:p:381-392
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    References listed on IDEAS

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    1. Asok Nanda & Moshe Shaked, 2001. "The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 853-864, December.
    2. Bon, Jean-Louis & Kalashnikov, Vladimir, 2001. "Some estimates of geometric sums," Statistics & Probability Letters, Elsevier, vol. 55(1), pages 89-97, November.
    3. Masaaki Kijima & Toshio Nakagawa, 1991. "A cumulative damage shock model with imperfect preventive maintenance," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(2), pages 145-156, April.
    4. Willmot, Gordon E. & Cai, Jun, 2001. "Aging and other distributional properties of discrete compound geometric distributions," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 361-379, June.
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    Cited by:

    1. Shaked, Moshe, 2007. "Stochastic comparisons of multivariate random sums in the Laplace transform order, with applications," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1339-1344, July.
    2. Pavlova, Kristina P. & Cai, Jun & Willmot, Gordon E., 2006. "The preservation of classes of discrete distributions under convolution and mixing," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 391-405, April.
    3. F. G. Badía & C. Sangüesa, 2015. "Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(5), pages 345-356, August.
    4. Li, Gang & Cheng, Kan & Jiang, Xiaoyue, 2006. "Negative ageing property of random sum," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 737-742, April.
    5. Sandhya E & Latha C M, 2019. "Compound Extended Geometric Distribution and Some of Its Properties," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 8(3), pages 1-96, November.

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