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Let $X_1,...,X_n$ be a random sample from some unknown probability density$f$ defined on a compact homogeneous manifold $\mathbf M$ of dimension $d \ge1$. Consider a 'needlet frame' $\{\phi_{j \eta}\}$ describing a localisedprojection onto the space of eigenfunctions of the Laplace operator on $\mathbfM$ with corresponding eigenvalues less than $2^{2j}$, as constructed in\cite{GP10}. We prove non-asymptotic concentration inequalities for the uniformdeviations of the linear needlet density estimator $f_n(j)$ obtained from anempirical estimate of the needlet projection $\sum_\eta \phi_{j \eta} \int f\phi_{j \eta}$ of $f$. We apply these results to construct risk-adaptiveestimators and nonasymptotic confidence bands for the unknown density $f$. Theconfidence bands are adaptive over classes of differentiable andH\"{older}-continuous functions on $\mathbf M$ that attain their H\"{o}lderexponents.

Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds