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.

中文翻译：

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$${{\mathbb {R}}}^+$$ 上的 Mellin–Meijer 核密度估计

非参数核密度估计是一个非常自然的过程，它简单地利用了卷积运算的平滑能力。然而，当要估计正变量的密度（边界问题，尾部的虚假颠簸）时，它的表现很差。因此，据称适用于 $\mathbb{R}^+$ 支持的密度的基本核估计器的各种扩展，例如使用 Gamma 或其他非对称核的那些，在文献中比比皆是。然而，这些并不是基于任何类似于卷积的有效平滑操作，这通常会导致不一致。相比之下，在本文中，$\mathbb{R}^+$ 支持的密度的核估计器是通过使用 Mellin 卷积定义的，这是 $\mathbb{R}^+$ 上通常卷积的自然模拟。从那里，一个非常透明的理论流动并导致与 Meijer 的 $G$ 函数密切相关的新型非对称内核。该“Mellin-Meijer-核密度估计器”的众多令人愉悦的特性在论文中得到了证明。它的逐点和 $L_2$ 一致性（具有最佳收敛速度）是为一大类密度建立的，包括在 0 处无界并在其右尾显示幂律衰减的密度。通过模拟和一些真实数据分析进一步研究了它的实际行为。包括在 0 处无界的密度，并在其右尾显示幂律衰减。通过模拟和一些真实数据分析进一步研究了它的实际行为。包括在 0 处无界的密度，并在其右尾显示幂律衰减。通过模拟和一些真实数据分析进一步研究了它的实际行为。