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Intrinsic Hölder classes of density functions on Riemannian manifolds and lower bounds to convergence rates

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  • Ki, Dohyeong
  • Park, Byeong U.

Abstract

We consider Hölder classes of density functions on Riemannian manifolds in intrinsic perspectives. We develop a theorem that links such Hölder classes on Riemannian manifolds to those on Euclidean spaces. Using the theorem, we derive lower bounds to Lp-convergence rates (1≤p≤∞) for the estimation of the underlying density on a Riemannian manifold.

Suggested Citation

  • Ki, Dohyeong & Park, Byeong U., 2021. "Intrinsic Hölder classes of density functions on Riemannian manifolds and lower bounds to convergence rates," Statistics & Probability Letters, Elsevier, vol. 169(C).
    • Handle: RePEc:eee:stapro:v:169:y:2021:i:c:s0167715220302625
    DOI: 10.1016/j.spl.2020.108959
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    References listed on IDEAS

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    1. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    2. Kim, Yoon Tae & Park, Hyun Suk, 2013. "Geometric structures arising from kernel density estimation on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 112-126.
    3. Ying Yuan & Hongtu Zhu & Weili Lin & J. S. Marron, 2012. "Local polynomial regression for symmetric positive definite matrices," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(4), pages 697-719, September.
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