Quantile regression via an MM algorithm

Quantile regression is an increasingly popular method for estimating the qu antiles of a distribution conditional on the values of covariates. Regressi on quantiles are robust against the influence of outliers and, taken severa l at a time, they give a more complete picture of the conditional distribut ion than a single estimate of the center. This article first presents an it erative algorithm for finding sample quantiles without sorting and then exp lores a generalization of the algorithm to nonlinear quantile regression. O ur quantile regression algorithm is termed an MM, or majorize-minimize, alg orithm because it entails majorizing the objective function by a quadratic function followed by minimizing that quadratic. The algorithm is conceptual ly simple and easy to code, and our numerical tests suggest that it is comp utationally competitive with a recent interior point algorithm for most pro blems.