The Annals of Statistics

Deciding the dimension of effective dimension reduction space for functional and high-dimensional data

Yehua Li and Tailen Hsing

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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 χ2 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.

Article information

Ann. Statist., Volume 38, Number 5 (2010), 3028-3062.

First available in Project Euclid: 30 August 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression
Secondary: 62G20: Asymptotic properties 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Adaptive Neyman test dimension reduction elliptically contoured distribution functional data analysis principal components


Li, Yehua; Hsing, Tailen. Deciding the dimension of effective dimension reduction space for functional and high-dimensional data. Ann. Statist. 38 (2010), no. 5, 3028--3062. doi:10.1214/10-AOS816.

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