Density estimation using Dirichlet kernels

We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison & 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. Dirichlet kernel estimators, which generalize the unidimensional Beta kernel estimator from Chen (1999), are free of boundary bias and non-negative everywhere on the simplex. We show that they achieve the optimal convergence rate $O(n^{-4/(d+4)})$ for the mean squared error and the mean integrated squared error, we prove their asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean absolute error.