Functional Partial Least-Squares: Adaptive Estimation and Inference

TL;DR

This paper introduces an adaptive functional PLS estimator for scalar-on-Hilbert space regression, achieving near-optimal convergence and providing new inference tools, with demonstrated effectiveness in simulations and real data analysis.

Contribution

It develops an adaptive early stopping rule for functional PLS and introduces a new test for local alternatives, advancing estimation and inference in ill-posed inverse problems.

Findings

Estimator attains nearly minimax-optimal rates.

Proposed test detects local alternatives at parametric rates.

Method performs well in simulations and real data applications.

Abstract

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