Convolution-smoothing based locally sparse estimation for functional quantile regression

Motivated by an application to study the impact of temperature, precipitation and irrigation on soybean yield, this article proposes a sparse semi-parametric functional quantile model. The model is called ``sparse'' because the functional coefficients are only nonzero in the local time region where the functional covariates have significant effects on the response under different quantile levels. To tackle the computational and theoretical challenges in optimizing the quantile loss function added with a concave penalty, we develop a novel Convolution-smoothing based Locally Sparse Estimation (CLoSE) method, to do three tasks in one step, including selecting significant functional covariates, identifying the nonzero region of functional coefficients to enhance the interpretability of the model and estimating the functional coefficients. We establish the functional oracle properties and simultaneous confidence bands for the estimated functional coefficients, along with the asymptotic normality for the estimated parameters. In addition, because it is difficult to estimate the conditional density function given the scalar and functional covariates, we propose the split wild bootstrap method to construct the confidence interval of the estimated parameters and simultaneous confidence band for the functional coefficients. We also establish the consistency of the split wild bootstrap method. The finite sample performance of the proposed CLoSE method is assessed with simulation studies. The proposed model and estimation procedure are also illustrated by identifying the active time regions when the daily temperature influences the soybean yield.