Gaussian Copula Function-on-Scalar Regression in Reproducing Kernel Hilbert Space

Abstract

This thesis proposes a novel Gaussian copula function-on-scalar regression, which is more flexible to characterize the relationship between functional or image response and scalar predictors and is able to relax the linear assumption in traditional function-on-scalar linear regression. Estimation and prediction of the proposed model are investigated: we develop the closed form for the estimator of coefficient functions in a reproducing kernel Hilbert space without the knowledge of marginal transformations; A valid prediction band is constructed via conformal prediction methods with minimal assumptions. Theoretically, we establish the optimal convergence rate on the estimation of coefficient functions and show that our proposed estimator achieves the minimax rate under both fixed and random designs. Simulations and real data analysis are conducted to assess the finite-sample performance.