Minimax properties of Dirichlet kernel density estimators
Abstract
This paper is concerned with the asymptotic behavior in $\beta$-H\"older spaces and under $L^p$ losses of a Dirichlet kernel density estimator introduced by Aitchison & Lauder (1985) and studied theoretically by Ouimet & Tolosana-Delgado (2021). It is shown that the estimator is minimax when $p \in [1, 3)$ and $\beta \in (0, 2]$, and that it is never minimax when $p \in [4, \infty)$ or $\beta \in (2, \infty)$. These results rectify in a minor way and, more importantly, extend to all dimensions those already reported in the univariate case by Bertin & Klutchnikoff (2011).
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