Abstract: The conventional method for functional quantile regression (FQR) is to fit the regression model for each quantile of interest separately. Therefore, the slope function of the regression, as a bivariate function of time and quantile, is estimated as a univariate function of time for each fixed quantile. However, there are several limitations to this conventional strategy. For example, it cannot guarantee the monotonicity of the conditional quantiles, nor can it control the smoothness of the slope estimator as a bivariate function. In this paper, we propose a new framework for FQR, in which we simultaneously fit the FQR model for multiple quantiles, with the help of a bivariate basis under some constraints, such that the estimated quantiles satisfy the monotonicity conditions and the smoothness of the slope estimator is controlled. The proposed estimator for the slope function is shown to be asymptotically consistent, and we establish its asymptotic normality. We use simulation to evaluate the finite-sample performance of the proposed method and compare it with that of the conventional method. We demonstrate the proposed method by analyzing the effects of daily temperature on bike rentals, and by investigating the relationship between children's growth history and their adult height.

Key words and phrases: Bivariate spline basis, functional data analysis, non-crossing quantiles.