Abstract

In this paper, the optimal estimation at a given point is considered for biased density in theory. A lower bound is first provided for all possible estimators over the local Hölder space. Using wavelet method, a linear wavelet estimator is constructed and turned out to be optimal by investigating its upper bound of pointwise convergence rate. To get the adaptivity, a nonlinear wavelet estimator is obtained and proved to be near-optimal within a logarithmic factor.

中文翻译：

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偏密度的最优小波估计

摘要

在本文中，给定点的最优估计在理论上被认为是有偏的密度。首先为局部 Hölder 空间上的所有可能估计器提供一个下界。使用小波方法，构造了一个线性小波估计器，并通过研究其逐点收敛速度的上界证明是最优的。为了获得自适应性，获得了一个非线性小波估计器，并证明它在对数因子内是接近最优的。