Optimal subsampling for linear quantile regression models

Abstract

Subsampling techniques are efficient methods for handling big data. Quite a few optimal sampling methods have been developed for parametric models in which the loss functions are differentiable with respect to parameters. However, they do not apply to quantile regression (QR) models as the involved check function is not differentiable. To circumvent the non‐differentiability problem, we consider directly estimating the linear QR coefficient by minimizing the Hansen–Hurwitz estimator of the usual loss function for QR. We establish the asymptotic normality of the resulting estimator under a generic sampling method, and then develop optimal subsampling methods for linear QR. In particular, we propose a one‐stage subsampling method, which depends only on the lengths of covariates, and a two‐stage subsampling method, which is a combination of the one‐stage sampling and the ideal optimal subsampling methods. Our simulation and real data based simulation studies show that the two recommended sampling methods always outperform simple random sampling in terms of mean square error, whether the linear QR model is valid or not.

Nearly Optimal Two-step Poisson Sampling and Empirical Likelihood Weighting Estimation for M-estimation with Big Data

“…Regarding the first issue, it is generally accepted that carefully designed sampling probabilities make unequal probability samplings more efficient than simple random or uniform sampling. Many researchers have developed efficient or optimal sampling plans for frequently encountered parametric statistical problems, including linear regression models (Ma et al, 2014), logistic regression (Fithian and Hastie, 2014;Wang et al, 2018;Wang, 2019), softmax regression (Yao et al , 2023), generalized linear models (Ai et al, 2021b), quantile regression (Ai et al, 2021a;Fan et al, 2021;Wang and Ma, 2021), and more general models (Shen et al, 2021;Yu et al, 2022).…”

Subdata Selection With a Large Number of Variables

“…The current literature on subdata selection is rapidly growing. Much of the relevant literature focuses on identifying subdata that yields precise estimates of parameters in a given statistical model, for example, for linear regression [see, 10,23,8,36,41], logistic regression [42,39,7], multinomial logistic regression [46], generalized linear models [13,36,1,50,53,16], quantile regression [40,2,12,33], and quasi-likelihood [50]. All of these methods assume a true underlying model.…”

Optimal subsampling for functional quantile regression

“…Later, Yao and Wang (2019) and Ai et al (2021b) extended the subsampling method to softmax regression and generalized linear models, respectively. Very recently, Wang and Ma (2021), Ai et al (2021a), Fan et al (2021), and Shao et al (2022) employed the optimal subsampling method to ordinary quantile regression, and Shao and Wang (2021) and Yuan et al (2022) developed the subsampling for composite quantile regression.…”