  Multiple-group data is widely used in genomic studies, finance, and social
science. This study investigates a block structure that consists of covariate
and response groups. It examines the block-selection problem of
high-dimensional models with group structures for both responses and
covariates, where both the number of blocks and the dimension within each block
are allowed to grow larger than the sample size. We propose a novel strategy
for detecting the block structure, which includes the block-selection model and
a non-zero block selector (NBS). We establish the uniform consistency of the
NBS and propose three estimators based on the NBS to enhance modeling
efficiency. We prove that the estimators achieve the oracle solution and show
that they are consistent, jointly asymptotically normal, and efficient in
modeling extremely high-dimensional data. Simulations generate complex data
settings and demonstrate the superiority of the proposed method. A gene-data
analysis also demonstrates its effectiveness.
