Minimax properties of Dirichlet kernel density estimators

TL;DR

This paper analyzes the asymptotic minimax properties of the Dirichlet kernel density estimator for compositional data, identifying conditions under which it achieves optimal rates and where it does not.

Contribution

It establishes the minimax rates for the Dirichlet kernel estimator in certain parameter regimes and clarifies its limitations outside those regimes.

Findings

Achieves minimax rate for specific (p, β) ranges

Cannot be minimax for p ≥ 4 or β > 2

Extends results to multivariate case and corrects earlier findings

Abstract

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.