Wavelets are particularly useful because of their natural adaptive ability to characterize data with intrinsically local properties. When the data contain outliers or come from a population with a heavy-tailed distribution, $L_{1}$ -estimation should obtain a better fit. In this paper, we propose a $L_{1}$ -wavelet method for nonparametric regression, and derive the asymptotic properties of the $L_{1}$ -wavelet estimator, including the Bahadur representation, the rate of convergence and asymptotic normality. The rate of convergence of it is comparable with the optimal convergence rate of the nonparametric estimation in nonparametric models, and it does not require the continuously differentiable conditions of a nonparametric function.

中文翻译：

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\（L_ {1} \）-小波方法的非参数回归渐近

小波之所以特别有用，是因为它们具有天然的自适应能力，可以表征具有固有局部属性的数据。当数据包含异常值或来自具有大量尾巴分布的总体时，$ L_ {1} $-估计应获得更好的拟合度。在本文中，我们提出了一种用于非参数回归的$ L_ {1} $小波方法，并推导了$ L_ {1} $小波估计量的渐近性质，包括Bahadur表示，收敛速度和渐近正态性。它的收敛速度可与非参数模型中非参数估计的最佳收敛速度相媲美，并且不需要非参数函数的可连续微分条件。