Statistical inference for the slope parameter in functional linear regression

Abstract

In this paper, we consider the linear regression model Y=SX+ε with functional regressors and responses. We develop new inference tools to quantify deviations of the true slope S from a hypothesized operator S0 with respect to the Hilbert–Schmidt norm ~S−S0~2, as well as the prediction error E‖SX−S0X‖2. Our analysis is applicable to functional time series and based on asymptotically pivotal statistics. This makes it particularly user-friendly, because it avoids the choice of tuning parameters inherent in long-run variance estimation or bootstrap of dependent data. We also discuss two sample problems as well as change point detection. Finite sample properties are investigated by means of a simulation study. Mathematically, our approach is based on a sequential version of the popular spectral cut-off estimator SˆN for S. We prove that (sequential) plug-in estimators of the deviation measures are N-consistent and satisfy weak invariance principles. These results rest on the smoothing effect of L2-norms, that we exploit by a new proof-technique, the smoothness shift, which has potential applications in other fields.