Abstract

We study the partially linear single index model in the quantile regression context. We

propose a class of locally efficient estimators which rely on a working regression error distribution

model. We show the robustness of our estimator with respect to the misspecified error model. We

further clarify the fundamental difference between the quantile regression with the corresponding

partially linear single index mean regression problem. Asymptotic properties of the estimators are

established rigorously. We further study the efficiency issue in this model. Numerical examples are

implemented to illustrate the finite sample properties of our estimators. A real data application has

been done to show our estimator’s good performance.

Key words and phrases: Locally efficient, Quantile regression, Semiparametric efficiency

Information

Preprint No.SS-2025-0456
Manuscript IDSS-2025-0456
Complete AuthorsRuimiao Luo, Seungchul Baek, Yanyuan Ma
Corresponding AuthorsYanyuan Ma
Emailsyzm63@psu.edu

References

  1. Harrison, D. and Rubinfeld, D. L. (1978). Hedonic price and the demand for clean air. Journal of Environmental Economics and Management 5, 81–102.
  2. He, X. M. and Shi, P. D. (1998). Monotone B-Spline smoothing. Journal of the American Statistical Association 93, 643–650.
  3. Horowitz, J. L. and Shi, S. (2005). Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100, 1238–1249.
  4. Kong, E. and Xia, Y. C. (2012). A single-index quantile regression model and its estimation. Econometric Theory 28, 730–768.
  5. Lee, S. (2003). Efficient semiparametric estimation of a partially linear quantile regression model. Econometric Theory 19, 1–31.
  6. Li, L. X., Zhu, L. P. and Zhu, L. X. (2011). Inference on primary parameter of interest with the aid of dimension reduction estimation. Journal of the Royal Statistical Society B 73, 59–80.
  7. Lv, Y. Z., Zhang, R. Q., Zhao, W. H. and Liu, J. C. (2015). Quantile regression and variable selection of partial linear single-index model. Ann Inst Stat Math 67, 375–409.
  8. Ma, S. J. and He, X. M. (2016). Inference for single-index quantile regression models with profile optimization. Annals of Statistics 44, 1234–1268.
  9. Ma, Y. Y. and Zhu, L. P. (2013a). Efficient Estimation in Sufficient Dimension Reduction. Annals of Statistics 41, 250–268.
  10. Ma, Y. Y. and Zhu, L. P. (2013b). Efficiency Loss Caused by Linearity Condition in Dimension Reduction. Biometrika 100, 371–383.
  11. Ma, Y. Y. and Zhu, L. P. (2013c). Doubly Robust and Efficient Estimators for Heteroscedastic Partially Linear
  12. Single-index Model Allowing High Dimensional Covariates. Journal of the Royal Statistical Society, Series B 75, 305–322.
  13. Ma, Y. Y. and Zhu, L. P. (2014). On Estimation Efficiency of the Central Mean Subspace. Journal of the Royal St ti ti l S i t S i B 76 885 901
  14. Newey, W. K. and Powell, J. L. (1990). Efficient estimation of linear and type I censored regression models under conditional quantile restrictions. Econometric Theory 6, 295–317.
  15. Sherwood, B. and Wang, L. (2016). Partially linear additive quantile regression in ultra-high dimension. Annals of Statistics 44, 288–317.
  16. Wu, W. Z., Yu, K. M. and Yu, Y. (2010). Single-index quantile regression. Journal of Multivariate Analysis 101, 1607–1621.
  17. Zhang, Y. K., Lian, H. and Yu, Y. (2017). Estimation and variable selection for quantile partially linear single-index models. Journal of Multivariate Analysis 162, 215–234.

Acknowledgments

Luo’s work was supported by China Scholarship Council and Shanxi Scholarship Council of

China(grants 2022-012).

Supplementary Materials

The supplementary materials provide the proofs of Proposition 1, Theorem 1, Theorem 2

and Theorem 3 and additional figures.

Supplementary materials are available for download.