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.
Citation
Harrie Hendriks. "Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions." Ann. Statist. 18 (2) 832 - 849, June, 1990. https://doi.org/10.1214/aos/1176347628
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