  The increased availability of massive data sets provides a unique opportunity
to discover subtle patterns in their distributions, but also imposes
overwhelming computational challenges. To fully utilize the information
contained in big data, we propose a two-step procedure: (i) estimate
conditional quantile functions at different levels in a parallel computing
environment; (ii) construct a conditional quantile regression process through
projection based on these estimated quantile curves. Our general quantile
regression framework covers both linear models with fixed or growing dimension
and series approximation models. We prove that the proposed procedure does not
sacrifice any statistical inferential accuracy provided that the number of
distributed computing units and quantile levels are chosen properly. In
particular, a sharp upper bound for the former and a sharp lower bound for the
latter are derived to capture the minimal computational cost from a statistical
perspective. As an important application, the statistical inference on
conditional distribution functions is considered. Moreover, we propose
computationally efficient approaches to conducting inference in the distributed
estimation setting described above. Those approaches directly utilize the
availability of estimators from sub-samples and can be carried out at almost no
additional computational cost. Simulations confirm our statistical inferential
theory.
