Penalized spline smoothing is a popular and flexible method of obtaining estimates in nonparametric regression but the classical least-squares criterion is highly susceptible to model deviations and atypical observations. Penalized spline estimation with a resistant loss function is a natural remedy, yet to this day the asymptotic properties of M-type penalized spline estimators have not been studied. We show in this paper that M-type penalized spline estimators achieve the same rates of convergence as their least-squares counterparts, even with auxiliary scale estimation. We further find theoretical justification for the use of a small number of knots relative to the sample size. We illustrate the benefits of M-type penalized splines in a Monte-Carlo study and two real-data examples, which contain atypical observations.

中文翻译：

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带有辅助尺度估计的 M 型惩罚样条

惩罚样条平滑是一种在非参数回归中获得估计值的流行且灵活的方法，但经典的最小二乘标准很容易受到模型偏差和非典型观察的影响。具有抗性损失函数的惩罚样条估计是一种自然疗法，但迄今为止，尚未研究 M 型惩罚样条估计器的渐近特性。我们在本文中表明，即使使用辅助尺度估计，M 型惩罚样条估计器也能实现与其最小二乘对应物相同的收敛速度。我们进一步找到了相对于样本大小使用少量结的理论依据。我们在 Monte-Carlo 研究和两个包含非典型观察的真实数据示例中说明了 M 型惩罚样条的好处。