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

11/22/2018

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by   Rida Benhaddou, et al.

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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.