Year: 2010
Type: article
Abstract: We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ϵ, where X is the unobserved random signal of interest and ϵ is an independent random error with distribution φ. Under weak assumptions on the decay of the Fourier transform of φ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators fn for the density f of X, where f : ℝ → 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 φ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if φ 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.
Source: The Annals of Statistics
DOI: 10.1214/10-aos836
Authors: Karim Lounici, Richard Nickl
Institution: University of Cambridge
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2015759564