Asymptotic properties of the normalized discrete associated-kernel estimator for probability mass function

TL;DR

This paper studies the asymptotic behavior of a normalized discrete associated-kernel estimator for probability mass functions, establishing its consistency and normality under certain conditions, with practical implications demonstrated through simulations.

Contribution

It provides new theoretical results on the asymptotic properties of normalized discrete associated-kernel estimators, including conditions for convergence and normality.

Findings

Normalized estimator converges in mean square to 1.

Estimator is consistent and asymptotically normal.

Simulations confirm theoretical asymptotic behavior.

Abstract

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.