The Annals of Statistics

Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions

Harrie Hendriks

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Abstract

Supposing a given collection $y_1, \cdots, y_N$ of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The main hypothesis is that the manifold is closed and that the distribution is (sufficiently) smooth. Under such a hypothesis a convergence arbitrarily close to the $N^{-1/2}$ rate is possible, both in the $L_2$ and the $L_\infty$ senses.

Article information

Source
Ann. Statist., Volume 18, Number 2 (1990), 832-849.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347628

Digital Object Identifier
doi:10.1214/aos/1176347628

Mathematical Reviews number (MathSciNet)
MR1056339

Zentralblatt MATH identifier
0711.62036

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 58G11 58G25

Keywords
Nonparametric density estimation $L_2$ convergence $L_\infty$ convergence closed manifolds homogeneous manifolds Laplace-Beltrami operator Fourier theory convergence of generalized zeta functions

Citation

Hendriks, Harrie. Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions. Ann. Statist. 18 (1990), no. 2, 832--849. doi:10.1214/aos/1176347628. https://projecteuclid.org/euclid.aos/1176347628


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