Mathematics – Statistics Theory
Scientific paper
2010-11-11
Annals of Statistics 2010, Vol. 38, No. 5, 3028-3062
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/10-AOS816 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/10-AOS816
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.
Hsing Tailen
Li Yehua
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