Interquantile Shrinkage in Regression Models

Abstract

Conventional analysis using quantile regression typically focuses on fitting the regression model at different quantiles separately. However, in situations where the quantile coefficients share some common feature, joint modeling of multiple quantiles to accommodate the commonality often leads to more efficient estimation. One example of common features is that a predictor may have a constant effect over one region of quantile levels but varying effects in other regions. To automatically perform estimation and detection of the interquantile commonality, we develop two penalization methods. When the quantile slope coefficients indeed do not change across quantile levels, the proposed methods will shrink the slopes toward constant and thus improve the estimation efficiency. We establish the oracle properties of the two proposed penalization methods. Through numerical investigations, we demonstrate that the proposed methods lead to estimations with competitive or higher efficiency than the standard quantile regression estimation in finite samples. Supplementary materials for the article are available online.

Key Words:

• Fused LASSO
• Noncrossing
• Oracle
• Quantile regression
• Smoothing
• Sup-norm

ACKNOWLEDGMENTS

The authors are grateful to the editor, an associate editor, and two anonymous referees for their valuable comments. Wang’s research was supported in part by NSF (National Science Foundation) grant DMS-1007420. Bondell’s research was supported in part by NSF grant DMS-1005612 and NIH (National Institute of Health) grant P01-CA-142538 and NSF CAREER Award DMS-1149355.