Parametric modelling of thresholds across scales in wavelet regression

We propose a parametric wavelet thresholding procedure for estimation in the 'function plus independent, identically distributed Gaussian noise' model. To reflect the decreasing sparsity of wavelet coefficients from finer to coarser scales, our thresholds also decrease. They retain the noise-free reconstruction property while being lower than the universal threshold, and are jointly parameterised by a single scalar parameter. We show that our estimator achieves near-optimal risk rates for the usual range of Besov spaces. We propose a crossvalidation technique for choosing the parameter of our procedure. A simulation study demonstrates very good performance of our estimator compared to other state-of-the-art techniques. We discuss an extension to non-Gaussian noise.