We present a discrete kernel estimator appropriate for estimating probability mass functions (p.m.f's) for integer data. Discrete kernel functions analogous to the Beta functions used as kernels in the continuous case are derived for the interior and for the boundary of the domain. An integer bandwidth is considered. Cross validation is used for bandwidth selection. The estimator was motivated by the need to characterize processes (e.g., mixtures of geometric distributions) with long tailed distributions with high mass near the origin, and integer arguments of the random variable. Monte Carlo comparisons with the Hall and Titterington [8](HT) estimator are offered. An application for estimating the p.m.f.'s of wet and dry spell lengths for a nonparamet-ric renewal model of daily rainfall is also presented. Other possible methods for obtaining discrete weight sequences are also presented.