Density estimation using Dirichlet kernels

In this paper, we introduce Dirichlet kernels for the estimation of multivariate densities supported on the $d$-dimensional simplex. These kernels generalize the beta kernels from Brown & Chen (1999), Chen (1999), Chen (2000), Bouezmarni & Rolin (2003), originally studied in the context of smoothing for regression curves. We prove various asymptotic properties for the estimator: bias, variance, mean squared error, mean integrated squared error, asymptotic normality and uniform strong consistency. In particular, the asymptotic normality and uniform strong consistency results are completely new, even for the case $d = 1$ (beta kernels). These new kernel smoothers can be used for density estimation of compositional data. The estimator is simple to use, free of boundary bias, allocates non-negative weights everywhere on the simplex, and achieves the optimal convergence rate of $n^{-4/(d+4)}$ for the mean integrated squared error.