Global uniform risk bounds for wavelet deconvolution estimators

Abstract

We consider the statistical deconvolution problem where one observes $n$ replications from the model $Y=X+\epsilon$, where $X$ is the unobserved random signal of interest and $\epsilon$ is an independent random error with distribution $\varphi$. Under weak assumptions on the decay of the Fourier transform of $\varphi ,$ we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators $f_n$ for the density $f$ of $X$, where $f:\mathbb{R}\to \mathbb{R}$ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of $f$ if the Fourier transform of $\varphi$ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if $\varphi$ decays polynomially. We also analyze the case where $f$ is a "supersmooth"/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density $f$.

Published in at http://dx.doi.org/10.1214/10-AOS836 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (10)

• 2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization
• Concentration around the mean for maxima of empirical processes
• Exponential and moment inequalities for U-statistics
• Confidence bands in density estimation
• Weighted uniform consistency of kernel density estimators
• Uniform limit theorems for wavelet density estimators
• Periodic boxcar deconvolution and diophantine approximation
• Functional deconvolution in a periodic setting: Uniform case
• Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections
• Sharp optimality for density deconvolution with dominating bias

Cited by in corpus (21)

• Gaussian approximation of suprema of empirical processes
• Anti-concentration and honest, adaptive confidence bands
• Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression
• On adaptive inference and confidence bands
• Bayesian inverse problems with non-conjugate priors
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$
• Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data
• Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds
• Adaptive quantile estimation in deconvolution with unknown error distribution
• A uniform central limit theorem and efficiency for deconvolution estimators
• Confidence bands for multivariate and time dependent inverse regression models
• Honest Confidence Sets in Nonparametric IV Regression and Other Ill-Posed Models
• Inference on covariance operators via concentration inequalities: k-sample tests, classification, and clustering via Rademacher complexities
• Inference based on Kotlarski's Identity
• Structural adaptive deconvolution under $L_p$-losses
• Uniform confidence bands for nonparametric errors-in-variables regression
• Uniform confidence bands in deconvolution with unknown error distribution
• Improved Rademacher symmetrization through a Wasserstein based measure of asymmetry
• Multiscale Methods for Shape Constraints in Deconvolution: Confidence Statements for Qualitative Features
• The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach
• Density Deconvolution with Non-Standard Error Distributions: Rates of Convergence and Adaptive Estimation