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.

翻译：M型平滑样条是一类广泛的样条估计器，既包含流行的最小二乘平滑样条，也包含对异常观测和模型误设具有更强稳健性的样条估计器。然而，现有的渐近理论仅涵盖基于光滑目标函数的平滑样条估计器，因此未能涵盖常用的稳健估计器，如分位数平滑样条和Huber型平滑样条。本文对此进行了系统性研究，仅假设目标函数的凸性，证明了最小二乘（超）收敛速率可推广到迄今尚未描述其渐近性质的M型估计器。我们进一步证明，辅助尺度估计可在比文献中更弱的假设条件下处理，并为导数建立了最优收敛速率——这是在最小二乘框架外首次获得此类结果。模拟研究和实际数据示例表明：在常规数据上，非光滑M型样条与最小二乘样条具有相当的竞争力；而在包含异常值的数据上，前者表现出更优越的性能。