Mellin-Meijer-kernel density estimation on R+
by Gery Geenens
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 kernel estimator allegedly suitable for R+-supported densities, such as those using 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 R+-supported densities is defined by making use of the Mellin convolution, the natural analogue of the usual convolution on R+. From there, a very transparent theory flows and leads to a class of asymmetric kernels strongly related to Meijer G-functions. Numerous pleasant properties of this "Mellin-Meijer-kernel density estimator" are presented. Its pointwise- and L2-consistency (with optimal rate of convergence) are 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.