标题： 扭曲小波和垂直阈值处理 显示英文标题

标题： Warped Wavelet and Vertical Thresholding

摘要： 设 $\{(X_i,Y_i)\}_{i\in \{1,..., n\}}$ 是来自随机设计回归模型 $Y=f(X)+\epsilon$ 的一组独立同分布样本，其中 $(X,Y)\in [0,1]\times [-M,M]$ 表示相关参数。在处理此类模型时，自然希望在 $L^2([0,1],G_X)$ 范数下实现适应性，其中 $G_X(\cdot)$ 表示设计变量 $X$ 的（已知）边缘分布。近年来，许多工作致力于构建在这种设定下具有适应性的估计量（参见，例如，[5,24,25,32]），但只有少数估计量附带有易于实施的计算方案。 我们提出了一组基于 Picard 和 Kerkyacharian [36] 最近引入的扭曲小波基以及一种考虑小波系数层次结构（跨尺度结构）的树状阈值规则的估计器。我们证明，如果回归函数属于由 $G_X(\cdot)$ 定义的一类逼近空间，则我们的方法是自适应的，并且以最优速率收敛到真实的回归函数。结果以类似于 [19] 中的方式表示为超额概率。 显示英文摘要

摘要： Let $\{(X_i,Y_i)\}_{i\in \{1,..., n\}}$ be an i.i.d. sample from the random design regression model $Y=f(X)+\epsilon$ with $(X,Y)\in [0,1]\times [-M,M]$. In dealing with such a model, adaptation is naturally to be intended in terms of $L^2([0,1],G_X)$ norm where $G_X(\cdot)$ denotes the (known) marginal distribution of the design variable $X$. Recently much work has been devoted to the construction of estimators that adapts in this setting (see, for example, [5,24,25,32]), but only a few of them come along with a easy--to--implement computational scheme. Here we propose a family of estimators based on the warped wavelet basis recently introduced by Picard and Kerkyacharian [36] and a tree-like thresholding rule that takes into account the hierarchical (across-scale) structure of the wavelet coefficients. We show that, if the regression function belongs to a certain class of approximation spaces defined in terms of $G_X(\cdot)$, then our procedure is adaptive and converge to the true regression function with an optimal rate. The results are stated in terms of excess probabilities as in [19].