Minimax properties of Dirichlet kernel density estimators

This paper is concerned with the asymptotic behavior in $\beta$-H\"older spaces and under $L^p$ losses of a Dirichlet kernel density estimator proposed by Aitchison & Lauder (1985) for the analysis of compositional data. In recent work, Ouimet & Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this nonparametric estimator. As a complement, it is shown here that for $p \in [1, 3)$ and $\beta \in (0, 2]$, the Aitchison--Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth, but that this estimator cannot be minimax when either $p \in [4, \infty)$ or $\beta \in (2, \infty)$. These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin & Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.