  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 nonconvex 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.
