Fused estimators of the central subspace in sufficient dimension reduction

R. Dennis Cook, Xin Zhang

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


When studying the regression of a univariate variable Y on a vector x of predictors, most existing sufficient dimension-reduction (SDR) methods require the construction of slices of Y to estimate moments of the conditional distribution of X given Y. But there is no widely accepted method for choosing the number of slices, while a poorly chosen slicing scheme may produce miserable results. We propose a novel and easily implemented fusing method that can mitigate the problem of choosing a slicing scheme and improve estimation efficiency at the same time. We develop two fused estimators-called FIRE and DIRE-based on an optimal inverse regression estimator. The asymptotic variance of FIRE is no larger than that of the original methods regardless of the choice of slicing scheme, while DIRE is less computational intense and more robust. Simulation studies show that the fused estimators perform effectively the same as or substantially better than the parent methods. Fused estimators based on other methods can be developed in parallel: fused sliced inverse regression (SIR), fused central solution space (CSS)-SIR, and fused likelihood-based method (LAD) are introduced briefly. Simulation studies of the fused CSS-SIR and fused LAD estimators show substantial gain over their parent methods. A real data example is also presented for illustration and comparison. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)815-827
Number of pages13
JournalJournal of the American Statistical Association
Issue number506
StatePublished - 2014

Bibliographical note

Funding Information:
R. Dennis Cook is Professor (E-mail: dennis@stat.umn.edu), and Xin Zhang is Ph.D. student (E-mail: zhan0648@umn.edu), School of Statistics, University of Minnesota, Minneapolis, MN 55455. The authors are grateful to Yuexiao Dong and Bing Li for providing them the codes for computing the CSS estimators, and grateful to Liping Zhu, Lixing Zhu, and Zhenghui Feng for the cumulative slicing estimation codes. The authors also thank the editor, the associate editor, and two referees for those helpful comments that led to significant improvements in this article. Research for this article was supported in part by grant DMS-1007547 from the U.S. National Science Foundation.

Publisher Copyright:
© 2014 American Statistical Association.


  • Cumulative mean estimation
  • Inverse regression estimator
  • Sliced average variance estimation
  • Sliced inverse regression


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