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Abstract:
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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 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 R+-supported densities is defined by making use of the Mellin convolution, the natural analogue on R+ of the usual convolution. From there, a very transparent theory flows and naturally 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. Uniform consistency and rates of convergence are obtained, and its practical behaviour is illustrated by simulations and some real data analyses.
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