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:本文考虑了 Aitchison 和 Lauder (1985年) 提出的用于分析组成数据的 Dirichlet 内核密度估计器在 $\ beta$- H\" older 空格中和在 $LP$ 损失 $LP$ 下, 由 Aitchison 和 Lauder (1985年) 提出用于分析组成数据的 Dirichlet 内核密度估计器的无症状行为。 在最近的工作中, Oumit 和 Tolosana- Delgado (2022年) 建立了该估测器的统一一致性和无症状正常性。 作为补充, 这里显示 Aitchison- Lauder 估计器可以在 $( p,\ betatatatatatata) 损失中达到一个最小值的最小值, 只要$( bitatatatatatatatatata) 值在最小值( 2) 或最小值( 美元) 直径( testimal\ a maxin) 中无法成为最小值( maxin) maximal maxin maxilate maxin maxilate yal) yal y expe expe exm expeal $(美元) a cal) a cutin (美元) a cutin (美元) a cutin) ex) a cuteb ex ex) a lexlate ex)。