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Activity Number: 35 - Special Session: Section on Nonparametric Statistics Student Paper Competition
Type: Contributed
Date/Time: Sunday, July 30, 2017 : 2:00 PM to 3:50 PM
Sponsor: Section on Nonparametric Statistics
Abstract #322637 View Presentation
Title: Optimal Penalized Function-On-Function Regression Under a Reproducing Kernel Hilbert Space Framework
Author(s): Xiaoxiao Sun* and Pang Du and Xiao Wang and Ping Ma
Companies: University of Georgia and Virginia Tech and Purdue University and University of Georgia
Keywords: Function-on-Function regression ; Representer Theorem ; Reproducing kernel Hilbert space ; Penalized least squares ; Minimax convergence rate

Many studies collect data with response and predictor variables both being functions of some covariate. Their common goal is to understand the relationship between these functional variables. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression.

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