张弦平滑法的推广
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，如条件均值或$β$ --给定$X$的$Y$的分位数，其中$0<β<1$。为了实现这一点，我们最小化的标准，例如，$$ \sum_{i=1}^n ρ(f(X_i) - Y_i) + λ\cdot \mathop TV\nolimits (f) $$在所有候选功能$f$。这里，$ρ$是取决于特定回归函数的凸函数，$\mathop {\rm TV}\nolimits (f)$代表$f$的总变差，而$λ>0$是某个调整参数。这个框架进一步扩展到包括二进制或泊松回归，并包括本地化的总变差处罚。后者需要构造适应于$f$的非齐次光滑的估计。对于一般的框架，我们开发的非迭代算法的最小化问题的解决方案是密切相关的绷紧弦算法（参见。Davies和Kovac，2001年）。此外，我们建立了当前设置和单调回归之间的联系，扩展了Mammen和van de Geer（1997）以前的工作。算法的考虑和数值例子补充了两个一致性的结果。
英文摘要：
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 $β$--quantile of $Y$ given $X$, where $0<β<1$. In order to achieve this we minimize criteria such as, for instance, $$ \sum_{i=1}^n ρ(f(X_i) - Y_i) + λ\cdot \mathop TV\nolimits (f) $$ among all candidate functions $f$. Here $ρ$ 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 $λ>0$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
