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 that 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.

众所周知，分位数回归 (QR) 的估计方程可以使用 EM 算法求解，其中 M 步通过加权最小二乘法计算，在 E 步计算的权重作为独立广义逆高斯的期望变量。此处利用这一事实来扩展 QR 以允许线性预测器中的随机效应。该设置中算法的收敛性是通过表明它是广义交替最小化 (GAM) 过程来建立的。EM 算法的另一个修改还允许我们将最近提出的用于平均回归模型中变量选择的方法适应 QR 设置。模拟表明，所得方法明显优于使用套索惩罚的 QR 模型中的变量选择。