  Quantile regression is a robust and practically useful way to efficiently
model quantile varying correlation and predict varied response quantiles of
interest. This article constructs and tests MM algorithms, which are simple to
code and have been suggested superior to some other prominent quantile
regression methods in nonregularized problems, in an array of quantile
regression settings including linear (modeling different quantile coefficients
both separately and simultaneously), nonparametric, regularized, and monotone
quantile regression. Applications to various real data sets and two simulation
studies comparing MM to existing tested methods have corroborated our
algorithms' effectiveness. We have made one key advance by generalizing our MM
algorithm to efficiently fit easy-to-predict-and-interpret parametric quantile
regression models for data sets exhibiting manifest complicated nonlinear
correlation patterns, which has not yet been covered by current literature to
the best of our knowledge.
