Statistical inference for the slope parameter in functional linear regression

In this paper, we consider the linear regression model Y = SX + epsilon with functional regressors and responses. We develop new inference tools to quantify deviations of the true slope S from a hypothesized operator S-0 with respect to the Hilbert-Schmidt norm vertical bar vertical bar vertical bar vertical bar S - S-0 vertical bar vertical bar vertical bar vertical bar(2), as well as the prediction error E vertical bar vertical bar vertical bar vertical bar SX - S0X vertical bar vertical bar vertical bar vertical bar(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 (S) over cap (N) for S. We prove that (sequential) plug-in estimators of the deviation measures are root N-consistent and satisfy weak invariance principles. These results rest on the smoothing effect of L-2-norms, that we exploit by a new proof-technique, the smoothness shift, which has potential applications in other fields.