  We propose a new inferential framework for constructing confidence regions
and testing hypotheses in statistical models specified by a system of high
dimensional estimating equations. We construct an influence function by
projecting the fitted estimating equations to a sparse direction obtained by
solving a large-scale linear program. Our main theoretical contribution is to
establish a unified Z-estimation theory of confidence regions for high
dimensional problems.
  Different from existing methods, all of which require the specification of
the likelihood or pseudo-likelihood, our framework is likelihood-free. As a
result, our approach provides valid inference for a broad class of high
dimensional constrained estimating equation problems, which are not covered by
existing methods.
  Such examples include, noisy compressed sensing, instrumental variable
regression, undirected graphical models, discriminant analysis and vector
autoregressive models. We present detailed theoretical results for all these
examples. Finally, we conduct thorough numerical simulations, and a real
dataset analysis to back up the developed theoretical results.
