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Rates of the Strong Uniform Consistency for the Kernel-Type Regression Function Estimators with General Kernels on Manifolds

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Abstract

In the present paper, we develop strong uniform consistency results for the generic kernel (including the kernel density estimator) on Riemannian manifolds with Riemann integrable kernels in order to accomplish these difficult tasks. The kernels of the Vapnik-Chervonenkis class that are commonly utilized in statistical problems are different to the isotropic kernels we address in this paper. Moreover, we show, in the same context, the uniform consistency for nonparametric inverse probability of censoring weighted (IPCW) estimators of the regression function under random censorship. As an application, we present the strong uniform consistency for estimators of the Nadaray-Watson type, which is of independent interest.

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Notes

  1. The kernel is defined as \(K(\mathbf{x},\mathbf{y}):=\Psi(||\mathbf{x}-\mathbf{y}||)\) for a ‘‘proper’’ function \(\Psi\) defined on \(\mathbb{R}_{\geq 0}\) for all \(\mathbf{x},\mathbf{y}\in\iota(M_{d})\).

  2. The kernel is defined as \(K(\mathbf{x},\mathbf{y}):=\Phi(\mathbf{x}-\mathbf{y})\) for a ‘‘proper’’ function \(\Phi\) defined on \(\mathbb{R}^{p}\) for all \(\mathbf{x},\mathbf{y}\in\iota(M_{d})\).

  3. The kernel \(K\) is defined on \(\mathbb{R}^{d}\times\mathbb{R}^{d}\) and cannot be defined on \(\mathbf{x}-\mathbf{y}\) for all \(\mathbf{x},\mathbf{y}\in\mathbb{R}^{d}\).

  4. The density estimators are given by

    $$f_{n}(\mathbf{x}):=r_{n}(1;\mathbf{x})=\frac{1}{nh_{n}^{d}}\sum_{i=1}^{n}K\left(\frac{||\iota(\mathbf{X}_{i})-\iota(\mathbf{x})||_{\mathbb{R}^{p}}}{h_{n}}\right)$$
    (1.6)

    or

    $$f_{n}(\mathbf{x}):=r_{n}(1;\mathbf{x})=\frac{1}{nh_{n}^{d}U_{\mathbf{x}}(0)}\sum_{i=1}^{n}K_{h_{n}}(\mathbf{x},\mathbf{X}_{i}).$$
    (1.7)

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ACKNOWLEDGEMENTS

The author would like to thank the Editor-in-Chief, an Associate-Editor, and two referees for their extremely helpful remarks, which resulted in a substantial improvement of the original form of the work and a presentation that was more sharply focused.

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    1. Université de technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), CS60319-60203, Compiègne Cedex, France

      Salim Bouzebda & Nourelhouda Taachouche

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    1. Salim Bouzebda
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    Correspondence to Salim Bouzebda or Nourelhouda Taachouche.

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    CREDIT AUTHOR STATEMENT

    Nourelhouda TAACHOUCHE: conceptualization, methodology, investigation, writing–original draft, writing–review, and editing.

    Salim BOUZEBDA: conceptualization, methodology, investigation, writing–original draft, writing–revie, and editing.

    Both authors contributed equally to this work.

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    Bouzebda, S., Taachouche, N. Rates of the Strong Uniform Consistency for the Kernel-Type Regression Function Estimators with General Kernels on Manifolds. Math. Meth. Stat. 32, 27–80 (2023). https://doi.org/10.3103/S1066530723010027

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    • DOI: https://doi.org/10.3103/S1066530723010027

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