A Non-Uniform Bound on the Poisson-Negative Binomial Relative Error

The Stein-Chen method is used to determine a non-uniform bound for the relative error of the negative binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution functions with mean nq = n(1 − p). In view of this bound, it is indicated that the Poisson cumulative distribution function with this mean can be used as an estimate of the negative binomial cumulative distribution function when q is sufficiently small.