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Minimax properties of Dirichlet kernel density estimators

Author

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  • Bertin, Karine
  • Genest, Christian
  • Klutchnikoff, Nicolas
  • Ouimet, Frédéric

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.

Suggested Citation

  • Bertin, Karine & Genest, Christian & Klutchnikoff, Nicolas & Ouimet, Frédéric, 2023. "Minimax properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:jmvana:v:195:y:2023:i:c:s0047259x23000040
    DOI: 10.1016/j.jmva.2023.105158
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    References listed on IDEAS

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