Smoothing combined estimating equations in quantile regression for longitudinal data

Quantile regression has become a powerful complement to the usual mean regression. A simple approach to use quantile regression in marginal analysis of longitudinal data is to assume working independence. However, this may incur potential efficiency loss. On the other hand, correctly specifying a working correlation in quantile regression can be difficult. We propose a new quantile regression model by combining multiple sets of unbiased estimating equations. This approach can account for correlations between the repeated measurements and produce more efficient estimates. Because the objective function is discrete and non-convex, we propose induced smoothing for fast and accurate computation of the parameter estimates, as well as their asymptotic covariance, using Newton-Raphson iteration. We further develop a robust quantile rank score test for hypothesis testing. We show that the resulting estimate is asymptotically normal and more efficient than the simple estimate using working independence. Extensive simulations and a real data analysis show the usefulness of the method.