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

Kernel density estimation is a nonparametric procedure making use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is estimated, due to boundary issues. So, various extensions of the kernel estimator allegedly suitable for $${\mathbb {R}}^+$$ R + -supported densities, such as those using asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation. By contrast, in this paper a kernel density estimator is defined through the Mellin convolution, the natural analogue on $${\mathbb {R}}^+$$ R + of the usual convolution. From there, a class of asymmetric kernels related to Meijer G-functions is suggested, and asymptotic properties of this ‘Mellin–Meijer kernel density estimator’ are presented. In particular, its pointwise- and $$L_2$$ L 2 -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.