Some asymptotic results for functional lineal regression

Abstract

We study the asymptotic behavior of the slop estimator in functional linear regression model with functional outputs. It turns out that expansions of analytic functions of covariance operator is a valuable tool in the theory of functional data. It this dissertation we apply this tool to obtain an upper bound for the integrated squared error (ISE) of the functional regression estimator for both random and fixed design. A lower bound is also discussed in this paper and obtained by van Trees inequality in the fixed design case. Our calculations are based on abstract Hilbert space, which generalizes the optimal rate provided in Hall and Horowitz (2007)