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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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