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.

中文翻译：

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卷积结构密度模型中的点明智小波估计

通过使用核方法，Lepski和Willer在2017年和2019年的卷积结构密度模型中建立了自适应的和最佳\（L ^ p \）风险估计。他们假设它们的密度函数在Nikol'skii空间中。基于他们的工作，我们首先使用线性小波估计器在同一模型中获得逐点最优估计。我们允许我们的密度处于局部各向异性的Hölder空间中。然后，使用数据驱动的方法来获得自适应且接近最佳的估计。最后，我们显示了获得适应性所必需的对数因子。