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 based on observations from its noisy convolution, and convolution kernel unknown. We derive the minimax lower bounds for the mean squared error assuming that belongs to certain Besov space and the kernel function 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.
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