Statistical inference for the slope parameter in functional linear regression

Abstract

In this paper, we consider the linear regression model $Y=SX+\mathit{\epsilon }$ 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 $~S-S_0{~}^2$, as well as the prediction error $\mathbb{E}\Vert SX-S_{0}X{\Vert }^2$. 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 ${\hat{S}}_N$ for S. We prove that (sequential) plug-in estimators of the deviation measures are $\sqrt{N}$-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.