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 $\phi$. Under weak assumptions on the decay of the Fourier
transform of $\phi,$ 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 $\phi$ decays exponentially and that a
corresponding result holds true for the hard thresholding wavelet estimator if
$\phi$ 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 $f$.

研究了基于小波去卷积的密度估计方法，给出了有界密度的小波去卷积密度估计方法的有限样本上确界和最小上确界风险下确界，并证明了它们在 Besov 球上是最优的，同时，还研究了线性估计器如何适应于未知光滑度的情况，波浪阈值小波估计器如何适应于多项式渐变误差下的问题，最后还给出了基于 Rademacher 进程的全局置信带的构造方法。