Regularization of wavelet approximations
Regularization of wavelet approximations
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DOI：
10.1198/016214501753208942
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发表时间：
2001-09-01
影响因子：
3.7
通讯作者：
Fan, JQ
中科院分区：
文献类型：
--
作者：
Antoniadis, A;Fan, JQ
在本文中，我们引入非线性正则小波估计非参数回归函数时，采样点不是均匀间隔。这种方法可以很容易地应用于许多其他统计背景。提出了各种新的惩罚函数。Donoho和Johnstone的硬阈值和软阈值估计是非线性正则小波估计的特殊成员。它们对应于一类惩罚最小二乘估计量的上下包络。给出了正则化估计具有阈值性质的罚函数的必要条件。对于一类罚函数，得到了Oracle不等式和泛阈值参数。建立了非线性正则小波估计的采样性质，并证明了其自适应极小极大性。为了有效地解决惩罚最小二乘问题，非线性正则Sobolev插值（NRSI）被提出作为初始估计，它被证明具有良好的采样特性。通过正则化一步估计进一步改进了NRSI，正则化一步估计是使用NRSI作为初始估计的惩罚最小二乘问题的一步估计。还引入了分级非凸性算法来处理惩罚最小二乘问题。新引入的方法说明了几个数值例子。
In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are specific members of nonlinear regularized wavelet estimators. They correspond to the lower and upper envelopes of a class of the penalized least squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax. To efficiently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by regularized one-step estimators, which are the one-step estimators of the penalized least squares problems using the NRSI as initial estimators. The graduated nonconvexity algorithm is also introduced to handle penalized least squares problems. The newly introduced approaches are illustrated by a few numerical examples.