Duality Behaviors of the Quantile Regression Model Estimation Problem
A vector of quantile regression model coefficients, also known as regression quantiles, is shown to be the solution to a parametric minimization problem. It can also be shown that the same model parameters are obtainable by solving a nonparametric dual linear program, and it is this feature of the quantile regression model estimation problem (QRMEP) that is of particular interest. Both the primal and dual linear programs of the QRMEP are shown to possess special structures. Provided certain model assumptions are met, the QRMEP also exhibits two unique properties. These properties, along with the duality behaviors of the problem, are exploited in order to extend two pivoting algorithms to the class of QRMEPs: a generalization of interval-linear programming (I-LP) and a long-step variant of the dual simplex method. For problems and/or models up to a certain size, these extensions are shown to perform well, computationally, against the classic dual simplex algorithm and interior-point methods.