TEST ( IF 1.3 ) Pub Date : 2021-07-13 , DOI: 10.1007/s11749-021-00782-y Ioannis Kalogridis 1
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.
中文翻译:
具有非平滑目标函数的 M 型平滑样条的渐近性
M 型平滑样条是一大类样条估计器,包括流行的最小二乘平滑样条,但也包括不太容易受到异常观察和模型错误指定的样条估计器。然而,可用的渐近理论仅涵盖基于平滑目标函数的平滑样条估计量,因此忽略了常用的抗性估计量,如分位数和 Huber 型平滑样条。我们在本文中提供了一般处理,并假设仅目标函数的凸性,表明最小二乘(超)收敛率可以扩展到 M 型估计器,其渐近特性迄今尚未描述。我们进一步表明,辅助尺度估计可以在比文献中发现的假设弱得多的假设下进行处理,并且我们为导数建立了最佳收敛率,这些导数在最小二乘框架之外尚未获得。模拟研究和真实数据示例说明了非平滑 M 型样条在常规数据上相对于最小二乘样条的竞争性能及其在包含异常的数据上的优越性能。