Variable selection using penalized estimation methods in quantile regression models is an important step in screening for relevant covariates. In this paper, we present a one-step estimation procedure for variable selection in sparse, linear additive quantile regression models, using the SCAD penalty. The main idea of the proposed procedure is that the usual L1-norm objective function in quantile regression estimation is replaced by a smooth parametric approximation of this function, via iterative least squares computations. We conduct a simulation study and a real data analysis to check the finite sample performance of the one-step estimator. The results reveal that the one-step quantile SCAD method identifies relevant variables efficiently even when the process under study has heavy tails or if the process is contaminated with outliers.
References
[1]
Abdelmalek, N. N. (1974). On the Discrete Linear L1 Approximation and L1 Solutions of Overdetermined Linear Equations. Journal of Approximation Theory, 11, 38-53. https://doi.org/10.1016/0021-9045(74)90037-9
[2]
Amin, M., Song, L., Thorlie, M. A., & Wang, X. (2015). SCAD-Penalized Quantile Regression for High-Dimensional Data Analysis and Variable Selection. Statistica Neerlandica, 69, 212-235. https://doi.org/10.1111/stan.12056
[3]
Chen, L., & Lee, S. (2023). Sparse Quantile Regression. Journal of Econometrics, 235, 2195-2217. https://doi.org/10.1016/j.jeconom.2023.02.014
[4]
Deaton, A., & Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge University Press. https://doi.org/10.1017/cbo9780511805653
[5]
Fan, J., & Li, R. (2001). Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties. Journal of the American Statistical Association, 96, 1348-1360. https://doi.org/10.1198/016214501753382273
[6]
Fan, J., & Lv, J. (2008). Sure Independence Screening for Ultrahigh Dimensional Feature Space. Journal of the Royal Statistical Society Series B: Statistical Methodology, 70, 849-911. https://doi.org/10.1111/j.1467-9868.2008.00674.x
[7]
Khan, D. M., Yaqoob, A., Iqbal, N., Wahid, A., Khalil, U., Khan, M. et al. (2019). Variable Selection via SCAD-Penalized Quantile Regression for High-Dimensional Count Data. IEEE Access, 7, 153205-153216. https://doi.org/10.1109/access.2019.2948278
[8]
Koenker, R., & Bassett, G. (1978). Regression Quantiles. Econometrica, 46, 33-50. https://doi.org/10.2307/1913643
[9]
Lee, E. R., Noh, H., & Park, B. U. (2014). Model Selection via Bayesian Information Criterion for Quantile Regression Models. Journal of the American Statistical Association, 109, 216-229. https://doi.org/10.1080/01621459.2013.836975
[10]
Mai, T. T. (2024). A Sparse Pac-Bayesian Approach for High-Dimensional Quantile Prediction. Statistics and Computing, 35, Article No. 93. https://doi.org/10.1007/s11222-025-10617-7
[11]
Muggeo, V. M. R., Sciandra, M., & Augugliaro, L. (2011). Quantile Regression via Iterative Least Squares Computations. Journal of Statistical Computation and Simulation, 82, 1557-1569. https://doi.org/10.1080/00949655.2011.583650
[12]
Noh, H., & Lee, E. R. (2014). Component Selection in Additive Quantile Regression Models. Journal of the Korean Statistical Society, 43, 439-452. https://doi.org/10.1016/j.jkss.2014.01.002
[13]
Sherwood, B., & Maidman, A. (2022). Additive Nonlinear Quantile Regression in Ultra-High Dimension. Journal of Machine Learning Research, 23, 1-47.
[14]
Sherwood, B., & Wang, L. (2016). Partially Linear Additive Quantile Regression in Ultra-High Dimension. The Annals of Statistics, 44, 288-317. https://doi.org/10.1214/15-aos1367
[15]
Sherwood, B., Li, S., & Maidman, A. (2025). rqPen: Penalized Quantile Regression. R Package Version 4.1.3. https://github.com/bssherwood/rqpen
[16]
Song, Y., Li, Z., & Fang, M. (2022). Robust Variable Selection Based on Penalized Composite Quantile Regression for High-Dimensional Single-Index Models. Mathematics, 10, Article 2000. https://doi.org/10.3390/math10122000
[17]
Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology, 58, 267-288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x
[18]
Wang, L., Wu, Y., & Li, R. (2012). Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension. Journal of the American Statistical Association, 107, 214-222. https://doi.org/10.1080/01621459.2012.656014
[19]
Wu, Y., & Liu, Y. (2009). Variable Selection in Quantile Regression. Statistica Sinica, 19, 801-817. https://www.jstor.org/stable/24308857
[20]
Zhao, P., & Yu, B. (2006). On Model Selection Consistency of Lasso. The Journal of Machine Learning Research, 7, 2541-2563.
[21]
Zou, H. (2006). The Adaptive Lasso and Its Oracle Properties. Journal of the American Statistical Association, 101, 1418-1429. https://doi.org/10.1198/016214506000000735
[22]
Zou, H., & Hastie, T. (2005). Regularization and Variable Selection via the Elastic Net. Journal of the Royal Statistical Society Series B: Statistical Methodology, 67, 301-320. https://doi.org/10.1111/j.1467-9868.2005.00503.x
[23]
Zou, H., & Li, R. (2008). One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models. The Annals of Statistics, 36, 1509-1533 https://doi.org/10.1214/009053607000000802
