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Nonparametric recursive regression estimation on Riemannian Manifolds

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  • Khardani, Salah
  • Yao, Anne Françoise

Abstract

The considerations of this paper are restricted to random variables with values on Riemannian manifolds M and hence we propose a geometric framework to estimate their recursive regression function. Suppose we are given observations (Xi,Yi)i=1⋯n, where Xi∈M and Yi∈R. In this work we define and study a new estimator of the regression function on Riemannian Manifold M. Precisely, we employ a recursive version of the Nadaraya–Watson estimator on Riemannian Manifolds. Under some assumptions in Riemannian Manifolds data analysis, we study the properties of a recursive family kernels regression. The bias, variance are computed explicitly.

Suggested Citation

  • Khardani, Salah & Yao, Anne Françoise, 2022. "Nonparametric recursive regression estimation on Riemannian Manifolds," Statistics & Probability Letters, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:stapro:v:182:y:2022:i:c:s0167715221002364
    DOI: 10.1016/j.spl.2021.109274
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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. Roussas, George G., 1990. "Nonparametric regression estimation under mixing conditions," Stochastic Processes and their Applications, Elsevier, vol. 36(1), pages 107-116, 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. 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.
    5. Guillermo Henry & Daniela Rodriguez, 2009. "Robust nonparametric regression on Riemannian manifolds," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(5), pages 611-628.
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