International audienceDiscrete kernel estimation of a probability mass function (p.m.f.), often mentioned in the literature, has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f . We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such as Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to crossvalidation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed
Discrete triangular distributions are introduced, in order to serve as kernels in the non-parametric estimation for probability mass function. They are locally symmetric around every point of estimation. Their variances depend on the smoothing bandwidth and establish a bridge between Dirac and discrete uniform distributions. The boundary bias related to the discrete triangular kernel estimator is solved through a modification of the kernel near the boundary. The mean integrated squared errors and then the optimal bandwidth are investigated. We also study the adequate bandwidth for excess zeros. The performance of the discrete triangular kernel estimator is illustrated using simulated count data. An application to count data from football is described and compared with a binomial kernel estimator.
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