Distributed inference for quantile regression processes

Abstract

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.

Quantile regression under memory constraint

“…For example, in a special case of quantile estimation (i.e., p = 0), it is straightforward to show that √ n| β ndc − β(τ )| → ∞ in probability when n/m 2 → ∞ (see Theorem D.1 in Appendix). Similar phenomenon occurs for general p, see Volgushev, Chao and Cheng (2018). In fact, the local QR estimators β QR,k are biased estimators with the bias O(1/m).…”

“…, T N ) for some aggregation function G(·)). In recent years, this DC framework has been widely adopted in distributed statistical inference (see, e.g., Li, Lin and Li (2013); Chen and Xie (2014); Battey et al (2018); Zhao, Cheng and Liu (2016); Shi, Lu and Song (2017); Banerjee, Durot and Sen (2018) ;Volgushev, Chao and Cheng (2018) and Section 2 for detailed descriptions).…”

On the unbiased asymptotic normality of quantile regression with fixed effects

“…Proof. The first four statements (11)-(14) follow from Theorem S.2.1 of Volgushev, Chao, and Cheng (2017). More precisely, note that in the notation of Volgushev, Chao, and Cheng (2017) under the assumptions (A1)-(A4) we have g N = 0, c N = 0, ξ m , m are constant.…”

“…A key insight in the proofs is a detailed analysis of the expected values of remainder terms in the classical Bahadur representation for QR, while previous approaches (including Kato, Galvao, and Montes-Rojas (2012) and Galvao and Wang (2015)) focused on the stochastic order of those remainder terms. A similar analysis was previously performed in Volgushev, Chao, and Cheng (2017) for general QR models with growing dimension under the assumption of independent and identically distributed observations. The proofs involve subtle empirical process arguments, and extending those results to settings with dependent data requires a substantial amount of work.…”

“…where again φ iT (Z it , Y it ) are i.i.d. and centered, R We conclude this section by remarking that the idea of analyzing expected values of remainder terms when aggregating QR from subsets was also used in Volgushev, Chao, and Cheng (2017). However, the setting we consider here is different from the latter paper since we also need to take into account the W iT which also depend on β i (τ ± d T ) in a non-smooth way.…”