Improving estimation efficiency in quantile regression with longitudinal data

In this paper, we consider two weighted estimators to improve estimation efficiency in quantile regression with longitudinal data. The first estimator is from weighted quantile regression, and the second estimator is from weighted composite quantile regression, where the weights in the second estimator are from the first estimator. Different from earlier literature, our weights are quantile adaptive, which borrow information from the intra-subject correlation of the conditional quantile scores, rather than the conditional least squares scores. The weight for each subject is obtained from smoothed empirical likelihood quantile estimator, where quadratic inference function method is used to model the inverse of the correlation matrix of the conditional quantile scores. Under some regularity conditions, we prove that the weighted estimators are more efficient than the standard quantile regression estimators with equal weights. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.