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

Rida Benhaddou; Qing Liu

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arXiv · 1811.10411

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

Rida Benhaddou·Qing Liu

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Abstract

In the present paper, we consider the estimation of a periodic two-dimensional function $f(\cdot,\cdot)$ based on observations from its noisy convolution, and convolution kernel $g(\cdot,\cdot)$ 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.

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BibTeX RIS

Rida Benhaddou, Qing Liu. 2018-11-22. Minimax adaptive wavelet estimator for the anisotropic functional deconvolution model with unknown kernel. https://arxiv.org/abs/1811.10411

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