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A collection of quantile curves provides a complete picture of conditional distributions. A properly centered and scaled version of estimated curves at various quantile levels gives rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. Applications of our general process convergence results include the construction of non-crossing quantile curves and the estimation of conditional distribution functions. As a result of independent interest, we obtain a series of Bahadur representations with exponential bounds for tail probabilities of all remainder terms. Bounds of this kind are potentially useful in analyzing statistical inference procedures under divide-and-conquer setup.MSC 2010 subject classifications: Primary 62F12, 62G20, 62G08.4. New bounds for local basis functions: Last but not the least, in Sections 2.2 and 5.2, we provide results for models with "local basis structure" (for instance B-splines). For such basis functions, we show that the conditions on model dimension can be relaxed from m 4 = o(n 1−ε ) [required by Theorem 12 of Belloni et al. (2016)] to m 2 (log n) 6 = o(n) in the case of B-splines [see the discussion below Assumption (B1)].Given the discussions above, we would also like to point out that Belloni et al. (2016) discuss other aspects such as bootstrap approximations which are not covered in our paper. In summary, both Belloni et al. (2016) and the present paper consider the same model setup, but focus on different aspects of the resulting theory, and none of the two papers is more general than the other.The rest of this paper is organized as follows. Section 2 presents the weak convergence of QRP under general series approximation framework. Section 3 discusses the QRP in quantile partial linear models. As an application of our weak convergence theory, Section 4 considers various functionals of the quantile regression process. A detailed discussion on our novel Bahadur representations is given in Section 5, and all proofs are deferred to the appendix.

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.

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