Discrete kernel smoothing is now gaining importance in nonparametric statistics. In this paper, we investigate some asymptotic properties of the normalized discrete associated-kernel estimator of a probability mass function. We show, under some regularity and non-restrictive assumptions on the associated-kernel, that the normalizing random variable converges in mean square to 1. We then derive the consistency and the asymptotic normality of the proposed estimator. Various families of discrete kernels already exhibited satisfy the conditions, including the refined CoM-Poisson which is underdispersed and of second-order. Finally, the first-order binomial kernel is discussed and, surprisingly, its normalized estimator has a suitable asymptotic behaviour through simulations.
翻译:在非参数统计中,分解内核的平滑现在越来越重要。在本文中,我们调查了概率质量函数的正常离散相关内核估计值的一些无症状性能。我们显示,根据对相关内核的某些常规和非限制性假设,正常随机变数在正方形到1之间会合。然后,我们得出了拟议估测器的一致性和无症状的正常性。不同的离散内核的多个家庭已经展示出来,满足了条件,包括精细的COM-Poisson, 其分解和第二顺序。最后,讨论了一阶二阶的二阶内核,令人惊讶的是,其正常的估测器通过模拟具有适合的无症状行为。