The Annals of Statistics

Distributed inference for quantile regression processes

Stanislav Volgushev, Shih-Kang Chao, and Guang Cheng

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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 subsamples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1634-1662.

Received: February 2017
Revised: March 2018
First available in Project Euclid: 13 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62F12: Asymptotic properties of estimators 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

B-spline estimation conditional distribution function distributed computing divide-and-conquer quantile regression process


Volgushev, Stanislav; Chao, Shih-Kang; Cheng, Guang. Distributed inference for quantile regression processes. Ann. Statist. 47 (2019), no. 3, 1634--1662. doi:10.1214/18-AOS1730.

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Supplemental materials

  • Supplement to “Distributed inference for quantile regression processes”. The supplement contains additional technical remarks, simulation results and all proofs.