Adaptive Functional Linear Regression

Theoretical results in the functional linear regression literature have so far focused on minimax estimation where smoothness parameters are assumed to be known and the estimators typically depend on these smoothness parameters. In this paper we consider adaptive estimation in functional linear regression. The goal is to construct a single data-driven procedure that achieves optimality results simultaneously over a collection of parameter spaces. Such an adaptive procedure automatically adjusts to the smoothness properties of the underlying slope and covariance functions. The main technical tools for the construction of the adaptive procedure are functional principal component analysis and block thresholding. The estimator of the slope function is shown to adaptively attain the optimal rate of convergence over a large collection of function spaces. Keywords: Adaptive estimation; Block thresholding; Eigenfunction; Eigenvalue; Functional data analysis; Minimax estimation; Principal components analysis; Rate of convergence; Slope function; Smoothing; Spectral decomposition. AMS 2000 Subject Classi…cation: Primary 62J05; secondary 62G20. 1Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. The research of Tony Cai was supported in part by NSF Grant DMS-0604954. 2Department of Statistics, Yale University, New Haven, CT 06511. The research of Harrison Zhou was supported in part by NSF Grants DMS-0645676.