A weighted linear quantile regression

Abstract

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.

Keywords:

• adaptive weights
• birth-weight
• conditional quantile
• goodness of fit
• high cholesterol level
• linear programming
• rate of convergence
• weighted loss function

AMS 2010 Subject Classifications:

• Primary: 62J05
• Secondary: 62F10

Funding

This research is supported by the Natural Science and Engineering Research Council of Canada.

Acknowledgements

We express our appreciation for the insightful comments of the referees and editor, which helped to improve the paper.