Improvements in the Poisson approximation of mixed Poisson distributions |
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| Institution: | 1. Research Institute for Basic Sciences, Jeju National University, Jeju, Republic of Korea;2. Department of Computer Science and Statistics, Jeju National University, Jeju, Republic of Korea;1. Department of Mathematics & Statistics, University of New Hampshire, Durham, NH 03824, USA;2. Department of Statistics, University of Missouri, Columbia, MC 65211, USA;1. Novosibirsk State University, Novosibirsk, Russia;2. Novosibirsk State Technical University, Novosibirsk, Russia;3. Novosibirsk State University of Economics and Management, Novosibirsk, Russia;1. The Dow Chemical Company, Core R&D Analytical Sciences, Midland, MI 48674, USA;2. The Dow Chemical Company, Organics, Polymers & Organometallics, Midland, MI 48674, USA;3. The Dow Chemical Company, Materials Sciences and Engineering, Midland, MI 48674, USA |
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| Abstract: | We consider the approximation of mixed Poisson distributions by Poisson laws and also by related finite signed measures of higher order. Upper bounds and asymptotic relations are given for several distances. Even in the case of the Poisson approximation with respect to the total variation distance, our bounds have better order than those given in the literature. In particular, our results hold under weaker moment conditions for the mixing random variable. As an example, we consider the approximation of the negative binomial distribution, which enables us to prove the sharpness of a constant in the upper bound of the total variation distance. The main tool is an integral formula for the difference of the counting densities of a Poisson distribution and a related finite signed measure. |
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