Minimax adaptive wavelet estimator for the anisotropic functional deconvolution model with unknown kernel

Abstract

In the present paper, we consider the estimation of a periodic two-dimensional function $f(·,·)$ based on observations from its noisy convolution, and convolution kernel $g(·,·)$ unknown. We derive the minimax lower bounds for the mean squared error assuming that f belongs to certain Besov space and the kernel function g satisfies some smoothness properties. We construct an adaptive hard-thresholding wavelet estimator that is asymptotically near-optimal within a logarithmic factor in a wide range of Besov balls. The proposed estimation algorithm implements a truncation to estimate the wavelet coefficients, in addition to the conventional hard-thresholds. A limited simulations study confirms theoretical claims of the paper.

Keywords:

• Functional deconvolution
• minimax convergence rate
• L2-risk
• blind deconvolution