  It is known that the estimating equations for quantile regression (QR) can be
solved using an EM algorithm in which the M-step is computed via weighted least
squares, with weights computed at the E-step as the expectation of independent
generalized inverse-Gaussian variables. This fact is exploited here to extend
QR to allow for random effects in the linear predictor. Convergence of the
algorithm in this setting is established by showing that it is a generalized
alternating minimization (GAM) procedure. Another modification of the EM
algorithm also allows us to adapt a recently proposed method for variable
selection in mean regression models to the QR setting. Simulations show the
resulting method significantly outperforms variable selection in QR models
using the lasso penalty. Applications to real data include a frailty QR
analysis of hospital stays, and variable selection for age at onset of lung
cancer and for riboflavin production rate using high-dimensional gene
expression arrays for prediction.
