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February 2022 Dimension reduction for functional data based on weak conditional moments
Bing Li, Jun Song
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Ann. Statist. 50(1): 107-128 (February 2022). DOI: 10.1214/21-AOS2091


We develop a general theory and estimation methods for functional linear sufficient dimension reduction, where both the predictor and the response can be random functions, or even vectors of functions. Unlike the existing dimension reduction methods, our approach does not rely on the estimation of conditional mean and conditional variance. Instead, it is based on a new statistical construction—the weak conditional expectation, which is based on Carleman operators and their inducing functions. Weak conditional expectation is a generalization of conditional expectation. Its key advantage is to replace the projection on to an L2-space—which defines conditional expectation—by projection on to an arbitrary Hilbert space, while still maintaining the unbiasedness of the related dimension reduction methods. This flexibility is particularly important for functional data, because attempting to estimate a full-fledged conditional mean or conditional variance by slicing or smoothing over the space of vector-valued functions may be inefficient due to the curse of dimensionality. We evaluated the performances of the our new methods by simulation and in several applied settings.

Funding Statement

The research of Bing Li is supported in part by the U.S. National Science Foundation Grant DMS-1713078.


The authors would like to thank three referees and an Associate Editor for their many useful and constructive comments and suggestions, which helped us greatly in improving this work.


Download Citation

Bing Li. Jun Song. "Dimension reduction for functional data based on weak conditional moments." Ann. Statist. 50 (1) 107 - 128, February 2022.


Received: 1 September 2019; Revised: 1 March 2021; Published: February 2022
First available in Project Euclid: 16 February 2022

Digital Object Identifier: 10.1214/21-AOS2091

Primary: 62G08
Secondary: 62B05 , 62H99

Keywords: Carleman operator , Function-on-function regression , weak average variance estimate , weak conditional moment , weak directional regression , weak inverse regression estimate

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 1 • February 2022
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