论文标题
具有非平滑目标函数的M型平滑花纹的渐近学
Asymptotics for M-type smoothing splines with non-smooth objective functions
论文作者
论文摘要
M型平滑样条是一类广泛的样条估计量，其中包括流行的最小二乘平滑样条，以及样条估计器，这些估计量不太容易受到观察和模型 - 密码指定的影响。但是，可用的渐近理论仅涵盖基于光滑目标函数的平滑样条估计器，因此遗漏了常用的抗性估计量，例如分位数和Huber型平滑键。我们在本文中提供了一般处理，并且仅假设目标函数的凸度表明，最小二乘（超级）收敛速率可以扩展到迄今未描述的渐近性能的M型估计量。我们进一步表明，辅助量表的估计值可以在明显弱的假设下处理，而在文献中的假设明显较弱，并且我们为衍生物建立了最佳的收敛速率，而这些衍生物的收敛速率尚未在最小二乘框架之外获得。一项仿真研究和真实数据示例说明了与常规数据最小二乘样条有关的非平滑M型花键的竞争性能，以及它们在包含异常情况的数据上的出色性能。
M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline but also spline estimators that are less susceptible to outlying observations and model-misspecification. However, available asymptotic theory only covers smoothing spline estimators based on smooth objective functions and consequently leaves out frequently used resistant estimators such as quantile and Huber-type smoothing splines. We provide a general treatment in this paper and, assuming only the convexity of the objective function, show that the least-squares (super-)convergence rates can be extended to M-type estimators whose asymptotic properties have not been hitherto described. We further show that auxiliary scale estimates may be handled under significantly weaker assumptions than those found in the literature and we establish optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework. A simulation study and a real-data example illustrate the competitive performance of non-smooth M-type splines in relation to the least-squares spline on regular data and their superior performance on data that contain anomalies.
