Functional Partial Least-Squares: Adaptive Estimation and Inference*

We study the linear regression model with a scalar response and a functional predictor, a canonical example of an ill-posed inverse problem. We show that the functional partial least-squares (PLS) estimator achieves convergence rates that are nearly minimax-optimal over a class of ellipsoids and propose an adaptive early-stopping procedure for selecting the number of PLS components. In addition, we develop a new test that detects parametric local alternatives. The test can be inverted to construct confidence sets for the functional slope parameter. Simulation results show that the estimator performs favorably relative to several existing methods, and that the proposed test has good power. We apply our methodology to evaluate the nonlinear effects of temperature on corn and soybean yields.