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Nonlinear estimation in anisotropic multi-index denoising. (English) Zbl 1010.62029

Summary: In the framework of denoising a function depending on a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of O.V. Lepski ’s method of adaptation [Theory Probab Appl. 38, No. 3, 454-466 (1990); translation from Teor. Veroyatn. Primen. 35, No. 3, 459-470 (1990; Zbl 0725.62075); Theory Probab. Appl. 36, No. 4, 682-697 (1991); translation from Teor. Veroyatn. Primen. 36, No. 4, 645-659 (1991; Zbl 0738.62045)], and roughly consists in choosing the “coarsest” bandwidth such that the estimated bias is negligible.
However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular, that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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