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Geometric structures arising from kernel density estimation on Riemannian manifolds

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  • Kim, Yoon Tae
  • Park, Hyun Suk

Abstract

Estimating the kernel density function of a random vector taking values on Riemannian manifolds is considered. We make use of the concept of exponential map in order to define the kernel density estimator. We study the asymptotic behavior of the kernel estimator which contains geometric quantities (i.e. the curvature tensor and its covariant derivatives). Under a Hölder class of functions defined on a Riemannian manifold with some global losses, the L2-minimax rate and its relative efficiency are obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:114:y:2013:i:c:p:112-126
    DOI: 10.1016/j.jmva.2012.07.006
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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. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    3. Hendriks, Harrie, 2003. "Application of fast spherical Fourier transform to density estimation," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 209-221, February.
    4. Bai, Z. D. & Rao, C. Radhakrishna & Zhao, L. C., 1988. "Kernel estimators of density function of directional data," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 24-39, October.
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    Cited by:

    1. 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).
    2. Federico Ferraccioli & Eleonora Arnone & Livio Finos & James O. Ramsay & Laura M. Sangalli, 2021. "Nonparametric density estimation over complicated domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 346-368, April.
    3. Berry, Tyrus & Sauer, Timothy, 2017. "Density estimation on manifolds with boundary," Computational Statistics & Data Analysis, Elsevier, vol. 107(C), pages 1-17.
    4. Asta, Dena Marie, 2021. "Kernel density estimation on symmetric spaces of non-compact type," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    5. Khardani, Salah & Yao, Anne Françoise, 2022. "Nonparametric recursive regression estimation on Riemannian Manifolds," Statistics & Probability Letters, Elsevier, vol. 182(C).
    6. Arnone, Eleonora & Ferraccioli, Federico & Pigolotti, Clara & Sangalli, Laura M., 2022. "A roughness penalty approach to estimate densities over two-dimensional manifolds," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).

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