Robust spatial scalar-on-function regression: A Fisher-consistent redescending M-estimation approach

2026-05-01

AI Summary

This study addresses estimation bias and numerical instability in spatial scalar-on-function regression caused by vertical outliers, high-leverage functional predictors, and strong spatial dependence. To tackle these challenges, the authors propose a unified robust estimation framework that, for the first time in spatial functional regression, incorporates Fisher-consistent redescending loss functions—such as Andrews’ sine and Danish—combined with robust functional principal component analysis and bias-corrected M-estimating equations. A hybrid IRLS-Newton algorithm is developed to simultaneously estimate regression coefficients, spatial dependence parameters, and scale parameters in a robust manner. Monte Carlo simulations demonstrate that the proposed method substantially outperforms classical and Huber-type approaches under data contamination. Empirical analysis of air quality data from France further confirms its superior predictive accuracy and stability in estimating spatial dependence.

Abstract

We develop a Fisher-consistent redescending robust estimator for the spatial scalar-on-function regression model, where a scalar response depends on both a functional predictor and a spatial autoregressive lag. Existing estimation procedures for this model are typically based on likelihood methods or monotone-loss robust M-estimators. They may be highly sensitive to vertical outliers, leverage points in the functional predictor, and numerical instability induced by strong spatial dependence. To address these issues, we propose a new estimation framework that first applies robust functional principal component analysis to obtain a contamination-resistant finite-dimensional representation of the functional predictor and then estimates the resulting spatial regression model through a bias-corrected system of M-estimating equations. The proposed method allows redescending loss functions, including Andrews' sine and Danish losses, and jointly estimates the regression coefficients, spatial dependence parameter, and scale parameter within a unified Fisher-consistent framework. For computation, we develop a hybrid IRLS-Newton algorithm that combines weighted least-squares updates for the regression parameters with a Newton-Raphson update for the spatial parameter. We establish Fisher consistency, consistency, asymptotic normality, and the asymptotic distribution of the reconstructed slope function. Monte Carlo experiments show that the proposed estimators remain competitive under clean data and substantially outperform classical and Huber-type robust competitors under contamination, particularly in severe outlier settings. An application to French air-quality data further demonstrates improved predictive performance and stable estimation of spatial dependence. Our method has been implemented in the fcsar R package.

Problem

• spatial scalar-on-function regression
• robust estimation
• vertical outliers
• leverage points
• spatial dependence

Innovation

• redescending M-estimation
• spatial scalar-on-function regression
• robust functional PCA
• Fisher consistency
• hybrid IRLS-Newton algorithm