On bounds in Poisson approximation for distributions of independent negative-binomial distributed random variables
Abstract
Using the Stein–Chen method some upper bounds in Poisson approximation for distributions of row-wise triangular arrays of independent negative-binomial distributed random variables are established in this note.
Poisson Approximation for Independent Geometric Random Variables
The Stein-Chen method is used to derive two formulas of uniform and non-uniform bounds on Poisson approximation for a sum of n independent geometric random variables. Application of these formulas is illustrated with the Poisson approximation to the negative binomial distribution.
Improvements in the Poisson approximation of mixed Poisson distributions
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.