Abstract

In this paper, we prove that functional sliced inverse regression (FSIR)

achieves the optimal (minimax) rate for estimating the central space in functional

sufficient dimension reduction problems. First, we provide a concentration inequality for the FSIR estimator of the covariance of the conditional mean. Based

on this inequality, we establish the root-n consistency of the FSIR estimator of

the image of covariance of the conditional mean.

Second, we apply the most

widely used truncated scheme to estimate the inverse of the covariance operator

and identify the truncation parameter that ensures that FSIR can achieve the

optimal minimax convergence rate for estimating the central space. Finally, we

conduct simulations to demonstrate the optimal choice of truncation parameter

and the estimation efficiency of FSIR. To the best of our knowledge, this is the

first paper to rigorously prove the minimax optimality of FSIR in estimating the

central space for multiple-index models and general Y (not necessarily discrete).

Information

Preprint No.SS-2023-0396
Manuscript IDSS-2023-0396
Complete AuthorsRui Chen, Songtao Tian, Dongming Huang, Qian Lin, Jun S. Liu
Corresponding AuthorsJun S. Liu
Emailsjliu@stat.harvard.edu

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Acknowledgments

Lin’s research was supported in part by the National Natural Science Foundation of China (Grant 92370122, 11971257). Huang’s research was sup-

ported in part by NUS Start-up Grant A-0004824-00-0 and Singapore Ministry of Education AcRF Tier 1 Grant A-8000466-00-00. Liu’s research was

supported in part by the National Science Foundation of the United States

Supplementary Materials

The online Supplementary Material includes the proofs for all the theoretical results in the paper.

Supplementary materials are available for download.