Partial functional linear quantile regression

This paper studies estimation in partial functional linear quantile regression in which the dependent variable is related to both a vector of finite length and a function-valued random variable as predictor variables. The slope function is estimated by the functional principal component basis. The asymptotic distribution of the estimator of the vector of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. It is showed that this rate is optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. The convergence rate of the mean squared prediction error for the proposed estimators is also be established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.