Abstract

This paper considers the asymptotic behavior in β-Hölder spaces, and under Lp losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison–Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever (p,β)∈[1,3)×(0,2] or (p,β)∈Ad, where Ad is a specific subset of [3,4)×(0,2] that depends on the dimension d of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either p∈[4,∞) or β∈(2,∞). These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.