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

TL;DR

This paper develops non-asymptotic concentration inequalities for needlet density estimators on compact homogeneous manifolds, enabling the construction of adaptive confidence bands for unknown densities.

Contribution

It introduces non-asymptotic concentration inequalities for needlet-based density estimators on manifolds, facilitating adaptive confidence bands and risk assessment.

Findings

Derived non-asymptotic concentration inequalities for needlet estimators

Constructed adaptive confidence bands for densities on manifolds

Demonstrated applicability to differentiable and Hölder continuous functions

Abstract

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 $d \ge 1$. Consider a 'needlet frame' $\{\phi_{j \eta}\}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $\mathbf M$ with corresponding eigenvalues less than $2^{2j}$, 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.