Discrete associated kernels method and extensions

Discrete 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 cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed.