Estimating Functional Quantile Regression Through an Eigen‐Sparse Envelope Model

Functional quantile regression performs well in many practical applications, especially when dealing with data with complex dependency structures. In this article, we propose a novel method that extends the envelope model to functional quantile regression to improve the efficiency. The proposed method uses an envelope structure to accommodate the eigen structure of the covariance operator for the predictor. The core idea is to establish a one‐to‐one relationship between the functional envelope quantile model and the multivariate envelope quantile model, enabling the estimation of the former by exploiting this relationship after the latter is estimated via the generalized method of moments. Theoretical properties of the proposed method are established. By comparing with standard functional quantile regression in simulations and real data applications, we demonstrate the significant improvement by the proposed method in terms of cross‐validated prediction error.