  We study theoretically, for the first time, the Dirichlet kernel estimator
introduced by Aitchison and Lauder (1985) for the estimation of multivariate
densities supported on the $d$-dimensional simplex. The simplex is an important
case as it is the natural domain of compositional data and has been neglected
in the literature on asymmetric kernels. The Dirichlet kernel estimator, which
generalizes the (non-modified) unidimensional Beta kernel estimator from Chen
(1999), is free of boundary bias and non-negative everywhere on the simplex. We
show that it achieves the optimal convergence rate $O(n^{-4/(d+4)})$ for the
mean squared error and the mean integrated squared error, we prove its
asymptotic normality and uniform strong consistency, and we also find an
asymptotic expression for the mean integrated absolute error. To illustrate the
Dirichlet kernel method and its favorable boundary properties, we present a
case study on minerals processing.
