  This paper proposes a general two directional simultaneous inference (TOSI)
framework for high-dimensional models with a manifest variable or latent
variable structure, for example, high-dimensional mean models, high-dimensional
sparse regression models, and high-dimensional latent factors models. TOSI
performs simultaneous inference on a set of parameters from two directions, one
to test whether the assumed zero parameters indeed are zeros and one to test
whether exist zeros in the parameter set of nonzeros. As a result, we can
exactly identify whether the parameters are zeros, thereby keeping the data
structure fully and parsimoniously expressed. We theoretically prove that the
proposed TOSI method asymptotically controls the Type I error at the
prespecified significance level and that the testing power converges to one.
Simulations are conducted to examine the performance of the proposed method in
finite sample situations and two real datasets are analyzed. The results show
that the TOSI method is more predictive and has more interpretable estimators
than existing methods.
