This article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

本文研究了在存在异常数据的情况下拟合稳健可加模型的 M 型估计器。允许加法模型中的组件具有不同程度的平滑度。我们引入了一类新的基于小波的鲁棒M型估计器，用于在这种非齐次加法模型中同时执行加法分量估计和变量选择。每个加性分量都通过小波基的截断级数展开来近似，从而可以将该方法应用于不一定是 2 的幂的非等间隔数据和样本大小。加性分量的稀疏性以及每个内小波系数的稀疏性组件（组），导致双水平组变量选择问题。在这个框架中，我们讨论稳健估计和变量选择。提出了一种由坐标下降型快速加速近端梯度算法和阈值化组成的两阶段计算算法。当使用非凸下降损失函数时，和适当的非凸惩罚函数在组级别，我们建立估计的最佳收敛率。我们证明了稀疏可加模型在弱兼容性条件下的变量选择一致性。理论结果辅以一些模拟和实际数据分析，以及与其他现有方法的比较。