P-splines quantile regression estimation in varying coefficient models

Quantile regression, as a generalization of median regression, has been widely used in statistical modeling. To allow for analyzing complex data situations, several flexible regression models have been introduced. Among these are the varying coefficient models, that differ from a classical linear regression model by the fact that the regression coefficients are no longer constant but functions that vary with the value taken by another variable, such as for example, time. In this paper, we study quantile regression in varying coefficient models for longitudinal data. The quantile function is modeled as a function of the covariates and the main task is to estimate the unknown regression coefficient functions. We approximate each coefficient function by means of P-splines. Theoretical properties of the estimators, such as rate of convergence and an asymptotic distribution are established. The estimation methodology requests solving an optimization problem that also involves a smoothing parameter. For a special case the optimization problem can be transformed into a linear programming problem for which then a Frisch-Newton interior point method is used, leading to a computationally fast and efficient procedure. Several data-driven choices of the smoothing parameters are briefly discussed, and their performances are illustrated in a simulation study. Some real data analysis demonstrates the use of the developed method.