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Criteria for the stochastic ordering of random sums, with actuarial applications. (English) Zbl 1003.60022
Let $${\mathbf M}=(M_1,\ldots,M_{n})$$ and $${\mathbf M'}=(M_1',\ldots,M_{n}')$$ be random vectors of integer-valued random variables and let $${\mathbf M}$$ and $${\mathbf M'}$$ have the same univariate marginals but different joint distributions. For given sequences $$(X_{1k}),\ldots,(X_{nk})$$ and $$(X_{1k}'),\ldots,(X_{nk}')$$ of positive random variables let us consider the sums $S_{M_{i}}=\sum_{k=1}^{M_{i}}X_{ik},\quad S_{M_i'}=\sum_{k=1}^{M_i'}X_{ik}', \qquad 1\leq i\leq n.$ The sequences $$(X_{ik})$$ and $$(X_{ik}')$$ are identically distributed and independent of $${\mathbf M}$$ and $${\mathbf M'}$$. The sequences $$(X_{ik})$$ and $$(X_{i'k})$$ are independent for $$i\neq i'$$, but for a fixed $$i$$, no assumption is made on the dependence structure between elements of any particular sequence $$(X_{ik})$$. The authors give definitions of the upper orthant, the lower orthant, the concordance, and the supermodular ordering and prove that vectors of sums $$(S_{M_1},\ldots,S_{M_{n}})$$ and $$(S_{M_1'}',\ldots,S_{M_{n}'}')$$ are stochastically ordered in just the same way as the corresponding vectors $${\mathbf M}$$ and $${\mathbf M'}$$. Some applications of the obtained results to the collective risk model in actuarial science in which a portfolio consists of $$n\geq 2$$ classes of business whose respective numbers of claims are random variables $$M_1,\ldots, M_{n}$$ are presented.

##### MSC:
 60E15 Inequalities; stochastic orderings 91B30 Risk theory, insurance (MSC2010)
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##### References:
  DOI: 10.1080/15326349708807420 · Zbl 0871.60015  DOI: 10.2143/AST.28.1.519079 · Zbl 1137.91484  DOI: 10.1214/aop/1176991777 · Zbl 0646.60018  DOI: 10.1093/biomet/65.1.141 · Zbl 0394.92021  Cook R. D., Journal of the Royal Statistical Society Series B 43 pp 210– (1981)  DOI: 10.1016/S0167-6687(99)00057-8 · Zbl 1103.91358  DOI: 10.2143/AST.26.2.563219  Frees E. W., North American Actuarial Journal 2 pp 1– (1998) · Zbl 1081.62564  DOI: 10.2307/3314660 · Zbl 0605.62049  DOI: 10.1016/S0167-6687(99)00002-5 · Zbl 0942.60008  Goovaerts M. J., Effective actuarial methods (1990)  DOI: 10.1016/S0167-6687(99)00007-4 · Zbl 0945.62109  Hutchinson T. P., Continuous bivariate distributions, emphasising applications (1990) · Zbl 1170.62330  DOI: 10.2307/1427721 · Zbl 0763.60006  DOI: 10.1016/0047-259X(90)90013-8 · Zbl 0741.62061  Joe H., Multivariate models and dependence concepts (1997) · Zbl 0990.62517  DOI: 10.1080/03610928308827274 · Zbl 0312.62008  DOI: 10.1214/aoms/1177699260 · Zbl 0146.40601  DOI: 10.2307/2307574 · Zbl 0050.28201  DOI: 10.2307/2289314 · Zbl 0683.62029  DOI: 10.1016/S0167-6687(97)00032-2 · Zbl 0894.90022  DOI: 10.1007/3-540-48236-9  Pellerey F., Scandinavian Actuarial Journal pp 38– (1997) · Zbl 0926.62102  DOI: 10.1006/jmva.1999.1833 · Zbl 0939.60090  Shaked M., Stochastic orders and their applications (1994) · Zbl 0806.62009  DOI: 10.1006/jmva.1997.1656 · Zbl 0883.60016  DOI: 10.1214/aop/1176994668 · Zbl 0459.62010  DOI: 10.2307/3215013 · Zbl 0899.60075
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