Quantile Regression via the EM Algorithm

Abstract

The three-parameter asymmetric Laplace distribution (ALD) has received increasing attention in the field of quantile regression due to an important feature between its location and asymmetric parameters. On the basis of the representation of the ALD as a normal-variance–mean mixture with an exponential mixing distribution, this article develops EM and generalized EM algorithms, respectively, for computing regression quantiles of linear and nonlinear regression models. It is interesting to show that the proposed EM algorithm and the MM (Majorization–Minimization) algorithm for quantile regressions are really the same in terms of computation, since the updating formula of them are the same. This provides a good example that connects the EM and MM algorithms. Simulation studies show that the EM algorithm can successfully recover the true parameters in quantile regressions.

Keywords:

• Asymmetric Laplace distribution
• Generalized EM algorithm
• Generalized inverse Gaussian distribution
• MM algorithm
• Normal-variance–mean mixture.

Mathematics Subject Classification:

• 62F86.

Acknowledgments

The first author is supported by the National Science Fund of China (71301099) and the Chinese Education Ministry Social Science Fund (12YJC790293). The second author is supported by the National Science Fund of China (71001061) and Innovation Program of Shanghai Municipal Education Commission (13ZS063). The third author is supported by the National Science Fund of China (71271221). We would like to thank the editor and two anonymous referees for numerous constructive comments that improved the quality of the paper.