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Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. (English) Zbl 0711.62036
This paper bases on results of L. Devroye and L. Györfi [Nonparametric density estimation. The $$L_ 1$$-view. (1985; Zbl 0546.62015)] and ideas of E. Giné [Ann. Stat. 3, 1243-1266 (1975; Zbl 0322.62058)]. The author assumed that the sample space is a closed or homogeneous m-dimensional Riemannian manifold and proposed a nonparametric density estimator of generalized orthonormal series type $$f^*$$ for the s times differentiable density f. He proved $$L_ 2$$ and $$L_{\infty}$$ convergence rates: $E(\| f-f^*\|^ 2_{L_ 2})\leq O(N^{m(2s+m)^{-1}}N^{-1}),$ and $E(\| f-f^*\|^ 2_{L_{\infty}})\leq O(N^{2m(2s+m)^{-1}}N^{-1}).$
Reviewer: S.Zwanzig

##### MSC:
 62G07 Density estimation 62G05 Nonparametric estimation 35P20 Asymptotic distributions of eigenvalues in context of PDEs 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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