Pursuit of dynamic structure in quantile additive models with longitudinal data

We consider quantile additive models with dynamic (time-varying) component functions. We allow some of the component functions to be non-dynamic, and show, as expected but technically nontrivially, that estimators of the non-dynamic functions have a faster convergence rate. A penalization-based method, called dynamic structure pursuit, is proposed to automatically identify these non-dynamic functions. Finally, in the sparse setting, a four-stage estimation procedure is proposed which first identifies the nonzero component functions and then applies the identification strategy of the non-dynamic functions. Theoretical and numerical results are provided to illustrate the performance of the estimators.