  High-dimensional statistical inference with general estimating equations are
challenging and remain less explored. In this paper, we study two problems in
the area: confidence set estimation for multiple components of the model
parameters, and model specifications test. For the first one, we propose to
construct a new set of estimating equations such that the impact from
estimating the high-dimensional nuisance parameters becomes asymptotically
negligible. The new construction enables us to estimate a valid confidence
region by empirical likelihood ratio. For the second one, we propose a test
statistic as the maximum of the marginal empirical likelihood ratios to
quantify data evidence against the model specification. Our theory establishes
the validity of the proposed empirical likelihood approaches, accommodating
over-identification and exponentially growing data dimensionality. The
numerical studies demonstrate promising performance and potential practical
benefits of the new methods.
