标题： 通过紧绷弦的平滑扩展 显示英文标题

标题： Extensions of smoothing via taut strings

摘要： 假设我们观察到独立的随机对$(X_1,Y_1)$,$(X_2,Y_2)$, ...,$(X_n,Y_n)$。我们的目标是估计回归函数，如给定$X$的$Y$的条件均值或$\beta$-分位数，其中$0<\beta <1$。 为了达到这个目的，我们最小化诸如$$ \sum_{i=1}^n \rho(f(X_i) - Y_i) + \lambda \cdot \mathop TV\nolimits (f) $$等准则，在所有候选函数$f$中进行选择。这里的$\rho$是一个依赖于我们所考虑的特定回归函数的凸函数，$\mathop {\rm TV}\nolimits (f)$表示$f$的总变差，而$\lambda >0$是某个调节参数。该框架进一步扩展以包括二元或泊松回归，并包括局部总变差惩罚。后者用于构建适应$f$非均匀平滑性的估计量。对于一般框架，我们开发了非迭代算法来解决最小化问题，这些算法与紧绷弦算法密切相关（参见 Davies 和 Kovac，2001）。此外，我们建立了当前设置与单调回归之间的联系，扩展了 Mammen 和 van de Geer（1997）之前的成果。算法考虑和数值例子由两个一致结果加以补充。 显示英文摘要

摘要： Suppose that we observe independent random pairs $(X_1,Y_1)$, $(X_2,Y_2)$, >..., $(X_n,Y_n)$. Our goal is to estimate regression functions such as the conditional mean or $\beta$--quantile of $Y$ given $X$, where $0<\beta <1$. In order to achieve this we minimize criteria such as, for instance, $$ \sum_{i=1}^n \rho(f(X_i) - Y_i) + \lambda \cdot \mathop TV\nolimits (f) $$ among all candidate functions $f$. Here $\rho$ is some convex function depending on the particular regression function we have in mind, $\mathop {\rm TV}\nolimits (f)$ stands for the total variation of $f$, and $\lambda >0$ is some tuning parameter. This framework is extended further to include binary or Poisson regression, and to include localized total variation penalties. The latter are needed to construct estimators adapting to inhomogeneous smoothness of $f$. For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac, 2001). Further we establish a connection between the present setting and monotone regression, extending previous work by Mammen and van de Geer (1997). The algorithmic considerations and numerical examples are complemented by two consistency results.