In high-dimensional data analysis, bi-level sparsity is often assumed when covariates function group-wisely and sparsity can appear either at the group level or within certain groups. In such cases, an ideal model should be able to encourage the bi-level variable selection consistently. Bi-level variable selection has become even more challenging when data have heavy-tailed distribution or outliers exist in random errors and covariates. In this paper, we study a framework of high-dimensional M-estimation for bi-level variable selection. This framework encourages bi-level sparsity through a computationally efficient two-stage procedure. In theory, we provide sufficient conditions under which our two-stage penalized M-estimator possesses simultaneous local estimation consistency and the bi-level variable selection consistency if certain non-convex penalty functions are used at the group level. Both our simulation studies and real data analysis demonstrate satisfactory finite sample performance of the proposed estimators under different irregular settings.

在高维数据分析中，当协变量按​​组运行时，通常假设双级稀疏性，并且稀疏性可以出现在组级别或某些组内。在这种情况下，理想的模型应该能够始终如一地鼓励双水平变量选择。当数据具有重尾分布或随机误差和协变量中存在异常值时，双层变量选择变得更具挑战性。在本文中，我们研究了用于双层变量选择的高维 M 估计框架。该框架通过计算高效的两阶段程序鼓励双层稀疏性。理论上，如果在组级别使用某些非凸惩罚函数，我们提供了充分条件，在该条件下，我们的两阶段惩罚 M 估计器同时具有局部估计一致性和双级变量选择一致性。我们的模拟研究和实际数据分析都证明了所提出的估计器在不同不规则设置下的令人满意的有限样本性能。