Point-Wise Wavelet Estimation in the Convolution Structure Density Model

By using a kernel method, Lepski and Willer establish adaptive and optimal L p risk estimations in the convolution structure density model in 2017 and 2019. They assume their density functions to be in a Nikol’skii space. Motivated by their work, we first use a linear wavelet estimator to obtain a point-wise optimal estimation in the same model. We allow our densities to be in a local and anisotropic Hölder space. Then a data driven method is used to obtain an adaptive and near-optimal estimation. Finally, we show the logarithmic factor necessary to get the adaptivity.