This paper considers the asymptotic behavior in $\beta$-H\"older spaces, and under $L^p$ 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 $(p,\beta) \in [1, 3) \times (0, 2]$ or $(p, \beta) \in \mathcal{A}_d$, where $\mathcal{A}_d$ is a specific subset of $[3, 4) \times (0, 2]$ that depends on the dimension $d$ of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either $p \in [4, \infty)$ or $\beta \in (2, \infty)$. These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.

翻译：本文研究Aitchison与Lauder(1985)针对成分数据分析所提出的Dirichlet核密度估计量在$\beta$-Hölder空间及$L^p$损失下的渐近行为。在近期工作中，Ouimet与Tolosana-Delgado(2022)建立了该估计量的均匀强相合性与渐近正态性。作为补充，本文证明：当$(p,\beta) \in [1, 3) \times (0, 2]$或$(p, \beta) \in \mathcal{A}_d$时（其中$\mathcal{A}_d$为$[3, 4) \times (0, 2]$中依赖于Dirichlet核维度$d$的特定子集），通过适当选择带宽，Aitchison-Lauder估计量可渐近达到极小极大速率。本文还证明当$p \in [4, \infty)$或$\beta \in (2, \infty)$时，该估计量无法达到极小极优性。这些结果不仅推广至多元情形，还以微小方式修正了Bertin与Klutchnikoff(2011)关于Beta核估计量极小极大性质的早期结论。