Title: Mellin-Meijer-kernel density estimation on $\mathbb{R}^+$,
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.