Linearized maximum rank correlation estimation when covariates are functional

This paper extends the linearized maximum rank correlation (LMRC) estimation proposed by Shen et al. (2023) to the setting where the covariate is a function. However, this extension is nontrivial due to the difficulty of inverting the covariance operator, which may raise the ill-posed inverse problem, for which we integrate the functional principal component analysis to the LMRC procedure. The proposed estimator is robust to outliers in response and computationally efficient. We establish the rate of convergence of the proposed estimator, which is minimax optimal under certain smoothness assumptions. Furthermore, we extend the proposed estimation procedure to handle discretely observed functional covariates, including both sparse and dense sampling designs, and establish the corresponding rate of convergence. Simulation studies demonstrate that the proposed estimators outperform the other existing methods for some examples. Finally, we apply our method to a real data to illustrate its usefulness.