Abstract:
|
We consider a class of nonlinear function on function regression models with multiple prediction curves. Compared to the linear function-on-function regression model which contains the sum of integrals of the products of the prediction curves and the corresponding coefficient kernel functions, we have the sum of integrals of unknown nonlinear functions of the individual prediction functions and corresponding time arguments. Inspired by the Karhunen-Loeve decomposition of the signal part of the response function, we provide expansions of the nonlinear functions which have the minimum prediction error in a large class of expansions. A generalized functional eigenvalue problem is proposed to estimate the expansions with penalties imposed to control the number of terms in the expansion and the smoothness of the estimated functions. Algorithms have been proposed to solving the penalized eigenvalue problem. Asymptotic theory for our estimation is provided.
|