Statistica Sinica 28 (2018), 2455-2468

ADAPTIVE FUNCTIONAL LINEAR REGRESSION VIA

FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS

AND BLOCK THRESHOLDING

T. Tony Cai ^{1}, Linjun Zhang ^{1} and Harrison H. Zhou ^{2}

Abstract: 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.

Key words and phrases: Adaptive estimation, block thresholding, eigenfunction, eigenvalue, functional data analysis, functional principal component analysis, minimax estimation, rate of convergence, slope function, smoothing, spectral decomposition.