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Deciding the dimension of effective dimension reduction space for functional and high-dimensional data. (English) Zbl 1200.62115
Summary: 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.

##### MSC:
 62M20 Inference from stochastic processes and prediction 62J05 Linear regression; mixed models 62H25 Factor analysis and principal components; correspondence analysis 62L10 Sequential statistical analysis 62G20 Asymptotic properties of nonparametric inference 46N30 Applications of functional analysis in probability theory and statistics 65C60 Computational problems in statistics (MSC2010)
##### Software:
fda (R); gss; RSIR
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