Optimal Prediction for Additive Function-on-Function Regression

TL;DR

This paper introduces an additive nonlinear functional regression model using RKHS, providing optimal estimation rates, computational solutions, and demonstrating its effectiveness through simulations and financial data analysis.

Contribution

It develops a novel additive function-on-function regression framework with optimal convergence rates and practical computational methods for complex nonlinear functional data modeling.

Findings

Achieved optimal prediction error rates.

Developed a computationally efficient approximation.

Validated the approach with financial data analysis.

Abstract

As with classic statistics, functional regression models are invaluable in the analysis of functional data. While there are now extensive tools with accompanying theory available for linear models, there is still a great deal of work to be done concerning nonlinear models for functional data. In this work we consider the Additive Function-on-Function Regression model, a type of nonlinear model that uses an additive relationship between the functional outcome and functional covariate. We present an estimation methodology built upon Reproducing Kernel Hilbert Spaces, and establish optimal rates of convergence for our estimates in terms of prediction error. We also discuss computational challenges that arise with such complex models, developing a representer theorem for our estimate as well as a more practical and computationally efficient approximation. Simulations and an application to Cumulative Intraday Returns around the 2008 financial crisis are also provided.