Functional Partial Least-Squares: Adaptive Estimation and Inference

TL;DR

This work studies functional linear regression with a Hilbert-space-valued predictor, framing the slope as an ill-posed inverse problem. It introduces functional PLS built on a Krylov subspace and establishes nearly minimax-optimal convergence rates under a source condition, with an adaptive early stopping rule that achieves rate-optimal estimation and prediction without knowing the degree of ill-posedness. The authors develop inference tools, including a test based on a fitted-moment statistic and corresponding confidence sets, enabling local alternatives and bootstrap-based calibration. Through extensive simulations and an empirical application to nonlinear temperature effects on US crop yields, the method demonstrates strong estimation performance, competitive prediction, and practical utility for climate-agriculture questions. Overall, the paper provides a comprehensive theory and practical toolkit for adaptive estimation and inference in functional PLS contexts.

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.

Figures (8)

• Figure 1: Distribution of estimation errors (left columns) and prediction errors (right columns) for Models 1–3. Each boxplot summarizes the distribution of errors across 5,000 Monte Carlo simulations.
• Figure 2: Accuracy of asymptotic approximations. The left panels display the empirical finite-sample distribution of the test statistic $T_n$ under $H_0$ (blue) overlaid with its asymptotic distribution (orange) for each of the three models. The right panels present the corresponding QQ-plots, comparing empirical quantiles of $T_n$ to the theoretical asymptotic quantiles.
• Figure 3: Simulated power curves of the test. Power curves for the three models are shown with $n=100$ (left columns) and $n=200$ (right columns). The solid blue line plots the empirical rejection probability against the scale factor $\delta$, while the red dashed line marks the 5% nominal significance level.
• Figure 4: Nonlinear relationship between temperature and crop yields fitted using functional PLS (red curve) and step function approximation (black dash).
• Figure 5: Adaptation effects in nonlinear relationship between temperature and crop yields.

Theorems & Definitions (36)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 3
• Remark 4