Sharp estimates in signed Poisson approximation of Poisson mixtures

Abstract

We give sharp estimates in total variation and certain kinds of stop-loss metrics in signed Poisson approximation of Poisson mixtures. We provide closed-form solutions to the problem of best choice of the Poisson parameter in simple Poisson approximation with respect to the total variation distance. The important special case of the negative binomial distribution is also discussed. To obtain our results, we apply a differential calculus based on different Taylor formulae for the Poisson process which allows us to give simple unified proofs.