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Compound Poisson approximation to convolutions of compound negative binomial variables. (English) Zbl 1320.60044
Summary: In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error bound using the method of exponents and it is compared with existing ones. It is known that Kerstan’s method [J. Kerstan, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 173–179 (1964; Zbl 0123.35403)] is more powerful in compound approximation problems. We employ Kerstan’s method to obtain better estimates, using higher-order approximations. These bounds are of higher-order accuracy and improve upon some of the known results in the literature. Finally, an interesting application to risk theory is discussed.

##### MSC:
 60E05 Probability distributions: general theory 60F05 Central limit and other weak theorems 60E15 Inequalities; stochastic orderings 91B30 Risk theory, insurance (MSC2010)
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