Estimation and Model Selection for Nonparametric Function-on-Function Regression

Regression models with a functional response and functional covariate have received significant attention recently. While various nonparametric and semiparametric models have been developed, there is an urgent need for model selection and diagnostic methods. In this article, we develop a unified framework for estimation and model selection in nonparametric function-on-function regression. We propose a general nonparametric functional regression model with the model space constructed through smoothing spline analysis of variance (SS ANOVA). The proposed model reduces to some of the existing models when selected components in the SS ANOVA decomposition are eliminated. We propose new estimation procedures under either <i>L</i>₁ or <i>L</i>₂ penalty and show that the combination of the SS ANOVA decomposition and <i>L</i>₁ penalty provides powerful tools for model selection and diagnostics. We establish consistency and convergence rates for estimates of the regression function and each component in its decomposition under both the <i>L</i>₁ and <i>L</i>₂ penalties. Simulation studies and real examples show that the proposed methods perform well. Technical details and additional simulation results are available in online supplementary materials.