Robust and smoothing variable selection for quantile regression models with longitudinal data

In this paper, we propose a penalized weighted quantile estimating equations (PWQEEs) method to obtain sparse, robust and efficient estimators for the quantile regression with longitudinal data. The PWQEE incorporates the within correlations in the longitudinal data by Gaussian copulas and can also down-weight the high leverage points in covariates to achieve double-robustness to both the non-normal distributed errors and the contaminated covariates. To overcome the obstacles of discontinuity of the PWQEE and nonconvex optimization, a local distribution smoothing method and the minimization–maximization algorithm are proposed. The asymptotic properties of the proposed method are also proved. Furthermore, finite sample performance of the PWQEE is illustrated by simulation studies and a real-data example.