- Series
- Job Candidate Talk
- Time
- Thursday, February 4, 2010 - 3:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Karim Lounici – University of Cambridge
- Organizer
- Yuri Bakhtin

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 where $\epsilon$ 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 nonasymptotic
confidence bands for the density $f$.