Mellin-Meijer-kernel density estimation on $\mathbb{R}^+$

TL;DR

This paper introduces a novel kernel density estimator for positive data using Mellin convolution, leading to consistent, asymptotically optimal estimation with practical validation through simulations and real data analysis.

Contribution

It develops a new Mellin-Meijer kernel density estimator based on Mellin convolution, addressing boundary issues and inconsistencies of previous methods for $ ext{R}^+$-supported densities.

Findings

Establishes pointwise and $L_2$-consistency with optimal convergence rates.

Demonstrates the estimator's effectiveness through simulations.

Validates practical performance with real data applications.

Abstract

Nonparametric kernel density estimation is a very natural procedure which simply makes use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is to be estimated (boundary issues, spurious bumps in the tail). So various extensions of the basic kernel estimator allegedly suitable for $\mathbb{R}^+$-supported densities, such as those using Gamma or other asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation analogous to the convolution, which typically leads to inconsistencies. By contrast, in this paper a kernel estimator for $\mathbb{R}^+$-supported densities is defined by making use of the Mellin convolution, the natural analogue of the usual convolution on $\mathbb{R}^+$. From there, a very transparent theory flows and leads to new type of asymmetric kernels strongly related to Meijer's $G$-functions. The numerous pleasant properties of this `Mellin-Meijer-kernel density estimator' are demonstrated in the paper. Its pointwise and $L_2$-consistency (with optimal rate of convergence) is established for a large class of densities, including densities unbounded at 0 and showing power-law decay in their right tail. Its practical behaviour is investigated further through simulations and some real data analyses.