Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
Résumé
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 a parts per thousand yen 1. Consider a 'needlet frame' describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 2(2j) , 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 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 Holder-continuous functions on M that attain their Holder exponents.