Penalized spline estimation with discrete difference penalties (P-splines) is a popular estimation method for semiparametric models, but the classical least-squares estimator is highly sensitive to deviations from its ideal model assumptions. To remedy this deficiency, a broad class of P-spline estimators based on general loss functions is introduced and studied. Robust estimators are obtained by well-chosen loss functions, such as the Huber or Tukey loss function. A preliminary scale estimator can also be included in the loss function. It is shown that this class of P-spline estimators enjoys the same optimal asymptotic properties as least-squares P-splines, thereby providing strong theoretical motivation for its use. The proposed estimators may be computed very efficiently through a simple adaptation of well-established iterative least squares algorithms and exhibit excellent performance even in finite samples, as evidenced by a numerical study and a real-data example.

具有离散差值惩罚的惩罚样条估计（P 样条）是半参数模型的流行估计方法，但经典的最小二乘估计器对其理想模型假设的偏差高度敏感。为了弥补这一缺陷，引入并研究了一类基于一般损失函数的 P 样条估计器。鲁棒估计量是通过精心选择的损失函数获得的，例如 Huber 或 Tukey 损失函数。损失函数中还可以包含初步规模估计器。结果表明，此类 P 样条估计器具有与最小二乘 P 样条相同的最优渐近性质所提出的估计量可以通过对完善的迭代最小二乘算法的简单调整来非常有效地计算，并且即使在有限样本中也表现出优异的性能，如数值研究和真实数据示例所证明的那样。