Sufficient dimension reduction via inverse regression: A minimum discrepancy approach

R. Dennis Cook, Liqiang Ni

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248 Scopus citations


A family of dimension-reduction methods, the inverse regression (IR) family, is developed by minimizing a quadratic objective function. An optimal member of this family, the inverse regression estimator (IRE), is proposed, along with inference methods and a computational algorithm. The IRE has at least three desirable properties: (1) Its estimated basis of the central dimension reduction subspace is asymptotically efficient, (2) its test statistic for dimension has an asymptotic chi-squared distribution, and (3) it provides a chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors. Current methods like sliced inverse regression belong to a suboptimal class of the IR family. Comparisons of these methods are reported through simulation studies. The approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in sliced average variance estimation.

Original languageEnglish (US)
Pages (from-to)410-428
Number of pages19
JournalJournal of the American Statistical Association
Issue number470
StatePublished - Jun 2005

Bibliographical note

Funding Information:
R. Dennis Cook is Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: [email protected]). Liqiang Ni is Assistant Professor, Department of Statistics and Actuarial Science, University of Central Florida, Orlando, FL 32816 (E-mail: [email protected]). This research was supported in part by National Science Foundation grants DMS-01-03983 and DMS-04-05360. The authors thank the editor for his helpful guidance and the referees for their critiques.


  • Inverse regression estimator
  • Sliced average variance estimation
  • Sliced inverse regression
  • Sufficient dimension reduction


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