Semiparametric and multiplicative bias correction techniques for second-order discrete kernels

In this article, we propose to estimate the probability mass function (pmf) of a discrete supported random variable by a semiparametric bias corrected method using discrete associated kernels. This method consists in applying a two-stage multiplicative bias correction (MBC) approach for the initial parametric model in order to improve the accuracy of the estimator measured in terms of the vanishing bias. Various properties of the resulting semiparametric MBC discrete associated kernel estimator are provided (bias, variance, and mean integrated squared error). The common cross-validation technique and the Kullback–Leibler divergence are adapted for bandwidth selection. Monte Carlo simulations and a real-data application for count data illustrate the performance of the semiparametric-MBC estimator.