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

TL;DR

This work analyzes the normalized discrete associated-kernel estimator for a probability mass function on a general discrete support $\mathbb{T}$. It proves that the normalizing constant $C_n$ converges in mean square to $1$ under mild second-order kernel conditions, enabling consistency and asymptotic normality results for the normalized estimator $\widehat{f}_n$. Under the joint assumptions (A1)-(A2) and $\sqrt{n}h_n\to 0$, pointwise convergence to $f(x)$ and asymptotic normality with variance $f(x)(1-f(x))$ are established; CMP kernels satisfy these properties and often outperform the first-order binomial kernel in finite samples. The paper also provides empirical illustrations via cross-validated bandwidths, demonstrating the practical utility of normalization and highlighting the favorable performance of the CMP kernel, while confirming that binomial normalization can be practically consistent as well. Overall, the results justify using normalized discrete associated-kernel estimators for nonparametric pmf estimation and guide kernel choice and bandwidth selection in practice.

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.

Figures (2)

• Figure 1: Empirical distributions of $\sqrt{n}\{\widehat{f}_n(x)-f_A(x)\}$ at $x=6$ over $N_{sim}=500$ independent simulations with $n = 500$ and $h_n=\{\!\sqrt{n}\log(n)\}^{-1}$ using CoM-Poisson (left) and binomial (right) kernels. The smoothed kernel density is displayed in full line, and the centered Gaussian density with the same variance is plotted in dotted line.
• Figure 2: Empirical frequency with its corresponding binomial and CoM-Poisson kernel smoothers of count dataset of insect pests on Hura trees with $n=51$; see, e.g., Senga17.

Theorems & Definitions (12)

• Definition 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 2.2: Consistency
• Corollary 2.3: Uniform consistency
• Theorem 2.4: Asymptotic normality
• Proposition 2.5
• proof : Proof of Proposition \ref{['prop: Cn']}
• proof : Proof of Theorem \ref{['ThConsist']}
• proof : Proof of Corollary \ref{['CorollUC']}