The general asymptotic distribution theory for the functional regression model in Ruymgaart et al. [Some asymptotic theory for functional regression and classification (2009) Texas Tech University] simplifies considerably if an extra assumption on the random regressor is made. In the special case where the regressor is a stochastic process on the unit interval, Johannes [Privileged communication (2008)] assumes the regressor to be stationary, in which case the eigenfunctions of their covariance operator turn out to be known, so that only the eigenvalues are to be estimated. In the present paper we will also assume the eigenvectors to be known, but within an abstract setting. The simplification mentioned above is due to the circumstance that the covariance operator of the regressor commutes with its estimator as it can be constructed under the current conditions. Moreover, it is now possible to test linear hypotheses for the regression parameter that correspond to linear subspaces spanned by a finite number of the known eigenvectors.
Digital Object Identifier: 10.1214/12-IMSCOLL921