Robust function-on-function interaction regression

TL;DR

This paper introduces a robust estimation method for function-on-function regression models with quadratic and interaction effects, effectively handling outliers and improving model reliability in complex functional data analysis.

Contribution

It proposes a novel robust estimation strategy using functional principal component decomposition and $ au$-estimators, along with a robust BIC and stepwise selection for model accuracy.

Findings

The method performs well in Monte Carlo simulations.

It demonstrates robustness against outliers in real data.

Asymptotic properties are theoretically validated.

Abstract

A function-on-function regression model with quadratic and interaction effects of the covariates provides a more flexible model. Despite several attempts to estimate the model's parameters, almost all existing estimation strategies are non-robust against outliers. Outliers in the quadratic and interaction effects may deteriorate the model structure more severely than their effects in the main effect. We propose a robust estimation strategy based on the robust functional principal component decomposition of the function-valued variables and $\tau$-estimator. The performance of the proposed method relies on the truncation parameters in the robust functional principal component decomposition of the function-valued variables. A robust Bayesian information criterion is used to determine the optimum truncation constants. A forward stepwise variable selection procedure is employed to determine relevant main, quadratic, and interaction effects to address a possible model misspecification. The finite-sample performance of the proposed method is investigated via a series of Monte-Carlo experiments. The proposed method's asymptotic consistency and influence function are also studied in the supplement, and its empirical performance is further investigated using a U.S. COVID-19 dataset.