Bootstrap Inference for Quantile-based Modal Regression

Abstract

In this article, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap methods for the conditional mode. We also extend the preceding analysis to the case where the dimension of the covariate vector is increasing with the sample size. Finally, we conduct simulation experiments and a real data analysis using the U.S. wage data to demonstrate the finite sample performance of our inference method. The supplemental materials include the wage dataset, R codes and an appendix containing proofs of the main results, additional simulation results, discussion of model misspecification and quantile crossing, and additional details of the numerical implementation.

Keywords:

• High-dimensional CLT
• Kernel smoothing
• Modal regression
• Pivotal bootstrap
• Quantile regression

Supplemental Materials

The supplemental materials contain the wage dataset, R codes and an appendix containing all the proofs, additional simulation results, discussion of model misspecification and quantile crossing, and additional details of the numerical implementation.

Acknowledgments

The authors thank the editor, an associate editor, and three anonymous referees for their careful review that helped improve upon the quality of the article.

Additional information

Funding

The research of Tao Zhang is partially supported by NSF grant DMS-1952306. The research of Kengo Kato is partially supported by NSF grants DM-1952306 and DMS-2014636.