  In this paper, we consider regression models with a Hilbert-space-valued
predictor and a scalar response, where the response depends on the predictor
only through a finite number of projections. The linear subspace spanned by
these projections is called the effective dimension reduction (EDR) space. To
determine the dimensionality of the EDR space, we focus on the leading
principal component scores of the predictor, and propose two sequential
$\chi^2$ testing procedures under the assumption that the predictor has an
elliptically contoured distribution. We further extend these procedures and
introduce a test that simultaneously takes into account a large number of
principal component scores. The proposed procedures are supported by theory,
validated by simulation studies, and illustrated by a real-data example. Our
methods and theory are applicable to functional data and high-dimensional
multivariate data.
