Abstract
Let X 1, . . . , X n be a random sample from some unknown probability density f defined on a compact homogeneous manifold M of dimension d ≥ 1. Consider a ‘needlet frame’ \({\{\phi_{j\eta}\}}\) describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 22j, as constructed in Geller and Pesenson (J Geom Anal 2011). We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f n (j) obtained from an empirical estimate of the needlet projection \({\sum_\eta \phi_{j \eta}\int f \phi_{j \eta}}\) 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 Hölder-continuous functions on M that attain their Hölder exponents.
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Kerkyacharian, G., Nickl, R. & Picard, D. Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Relat. Fields 153, 363–404 (2012). https://doi.org/10.1007/s00440-011-0348-5
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DOI: https://doi.org/10.1007/s00440-011-0348-5