Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions
The Annals of Statistics · 1990 · Volume 18 · Issue 2
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.
Keywords
$L_\infty$ convergence, $L_2$ convergence, 35P20, 58G11, 58G25, 62G05, closed manifolds, convergence of generalized zeta functions, Fourier theory, homogeneous manifolds, Laplace-Beltrami operator, Nonparametric density estimation
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