Optimal subsampling for quantile regression in big data

Abstract

Summary We investigate optimal subsampling for quantile regression. We derive the asymptotic distribution of a general subsampling estimator and then derive two versions of optimal subsampling probabilities. One version minimizes the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator and the other minimizes that of the original parameter estimator. The former does not depend on the densities of the responses given covariates and is easy to implement. Algorithms based on optimal subsampling probabilities are proposed and asymptotic distributions, and the asymptotic optimality of the resulting estimators are established. Furthermore, we propose an iterative subsampling procedure based on the optimal subsampling probabilities in the linearly transformed parameter estimation which has great scalability to utilize available computational resources. In addition, this procedure yields standard errors for parameter estimators without estimating the densities of the responses given the covariates. We provide numerical examples based on both simulated and real data to illustrate the proposed method.

Optimal subsampling for linear quantile regression models

“…The main strategy we adopted was to study the asymptotic properties of the Hansen–Hurwitz estimator for the unknown parameter directly from the optimization problem instead of the accompanying estimating equation. In the process of revising our manuscript, we have noticed that Wang & Ma (2020) and Ai et al (2020a) obtain similar results on optimal subsampling for QR in the context of big data. Although we focus only on the usual linear QR models in this article, this strategy applies to any parametric problem involving a non‐differentiable loss function, such as composite QR (Zou & Yuan, 2008) and expectile regression (Newey & Powell, 1987).…”

A Review on Optimal Subsampling Methods for Massive Datasets

“…This algorithm is stated in Algorithm 7. It has shown that the rate of convergence of the final estimator is (n 1 R) −1/2 in Wang and Ma (2020). From the result, we know that, at 5% significance level, the seventh covariate (Euclidian distance) is not significant to the model and all others are significant.…”

“…The adaptive optimal subsampling algorithm for quantile regression was discussed in Wang and Ma (2020). The quantile regression estimates a specified quantile of the response variable conditional on the covariate variable, and has form q τ (y i |x i ) = x T i β, where τ represents that the τ -th quantile of y i given x i is measured.…”

“…This method was named as optimal subsampling methods motivated from the A-optimality criterion (OSMAC), and was improved in Wang (2019b) by adopting unweighted target functions for subsamples and Poisson subsampling. In addition to logistic regression, OSMAC was investigated to include softmax regression (Yao and Wang, 2018), generalized linear models (Ai et al, 2019), quantile regression (Wang and Ma, 2020) and quasilikelihood (Yu et al, 2020). This article aims at introducing the optimal subsampling method and illustrates its practical implements in R (R Core Team, 2020) with the following real data examples.…”

Sampling‐based estimation for massive survival data with additive hazards model

“…Wang et al 9 provided an information‐based optimal subdata selection approach in the context of linear models. Wang and Ma 10 investigated optimal subsampling for quantile regression. Zhang and Wang 11 proposed a distributed subsampling procedure for big data linear models.…”