Quantile regression via an MM algorithm

Quantile regression is an increasingly popular method for estimating the quantiles of a distribution conditional on the values of covariates. Regression quantiles are robust against the influence of outliers and, taken several at a time, they give a more complete picture of the conditional distribution than a single estimate of the center. This article first presents an iterative algorithm for finding sample quantiles without sorting and then explores a generalization of the algorithm to nonlinear quantile regression. Our quantile regression algorithm is termed an MM, or majorize-minimize, algorithm because it entails majorizing the objective function by a quadratic function followed by minimizing that quadratic. The algorithm is conceptually simple and easy to code, and our numerical tests suggest that it is computationally competitive with a recent interior point algorithm for most problems.