Abstract: For functional regression models with functional responses, we propose a nonparametric random-effects model using Gaussian process priors. The proposed model captures the heterogeneity nonlinearly and the covariance structure nonparametrically, enabling longitudinal studies of functional data. The model also has a flexible form of mean structure. We develop a procedure to estimate the unknown parameters and calculate the random effects nonparametrically. The procedure uses a penalized least squares regression and a maximum a posterior estimate, yielding a more accurate prediction. The statistical theory is discussed, including information consistency. Simulation studies and two real-data examples show that the proposed method performs well.

Key words and phrases: Functional linear model, function-on-function regression model, Gaussian process priors, nonlinear random effects.