An improved bound for negative binomial approximation with z-functions

In this article, we use Stein's method together with z-functions to give an improved bound for the total variation distance between the distribution of a non-negative integer-valued random variable X and the negative binomial distribution with parameters r∈R+ and p=1−q∈(0,1), where rqp is equal to the mean of X, E(X). The improved bound is sharper than that mentioned in Teerapabolarn and Boondirek (2010). We give three examples of the negative binomial approximation to the distribution of X concerning the negative hypergeometric, Pólya and negative Pólya distributions.