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

Let $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ be a random sample from some unknown probability density $f$ defined on a compact homogeneous manifold $\mathbf{M}$ of dimension $d \ge 1$\mathrm{.}Consider{a}^{\mathrm{\prime }}needletfram{e}^{\mathrm{\prime }}${\phi_{j \eta}}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $\mathbf M$withcorrespondingeigenvalueslessthan$2^{2j}$, as constructed in \cite{GP10}. 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 H"{older}-continuous functions on $\mathbf{M}$ that attain their H"{o}lder exponents.