A non-uniform bound on binomial approximation with w-functions

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We use Stein’s method and w-functions to determine a non-uniform bound for approximating the distribution of a non-negative integer-valued random variable X by a binomial distribution with parameters and where Additionally, we also give a new improvement of non-uniform bound for the distance between the cumulative distribution function of X and a binomial cumulative distribution function. For applications, we use the obtained result to approximate the hypergeometric, negative hypergeometric, Pólya and beta binomial distributions.