2011/02/11 by Gerard Kerkyacharian, Richard Nickl, Dominique Picard · 29 citations
Mathematics · #Density estimation #Differentiable function #Dimension (graph theory) #Eigenfunction #Eigenvalues and eigenvectors #Estimator #Geometry and complex manifolds #Laplace operator #Mathematical Analysis and Transform Methods #Probability density function #Projection (relational algebra) #Stochastic processes and statistical mechanics #math.ST #msc:42C40 #msc:60E15 #msc:62G07 #stat.TH
paper · pdf · doi:10.1007/s00440-011-0348-5
published in Probability Theory and Related Fields 153(1-2), 363-404 (Springer Science+Business Media) · Probability Theory and Related Fields, to appear
arxiv created 2011/02/11 · openalex publication_date 2011/03/17 · arxiv updated 2012/08/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let X1,...,Xn be a random sample from some unknown probability density f defined on a compact homogeneous manifold \mathbf M of dimension d ≥ 1. Consider a 'needlet frame' \ϕj η\ describing a localised projection onto the space of eigenfunctions of the Laplace operator on \mathbf M with corresponding eigenvalues less than 22j, as constructed in \citeGP10. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator fn(j) obtained from an empirical estimate of the needlet projection ∑ηϕj η ∫ f ϕj η of f. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density f. The confidence bands are adaptive over classes of differentiable and H"older-continuous functions on \mathbf M that attain their Hölder exponents.