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.
Publié le : 1990-06-14
Classification:
Nonparametric density estimation,
$L_2$ convergence,
$L_\infty$ convergence,
closed manifolds,
homogeneous manifolds,
Laplace-Beltrami operator,
Fourier theory,
convergence of generalized zeta functions,
62G05,
35P20,
58G11,
58G25
@article{1176347628,
author = {Hendriks, Harrie},
title = {Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions},
journal = {Ann. Statist.},
volume = {18},
number = {1},
year = {1990},
pages = { 832-849},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347628}
}
Hendriks, Harrie. Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions. Ann. Statist., Tome 18 (1990) no. 1, pp. 832-849. http://gdmltest.u-ga.fr/item/1176347628/