Sufficient dimension reduction via inverse regression: A minimum discrepancy approach

R. Dennis Cook; Liqiang Ni

doi:10.1198/016214504000001501

Sufficient dimension reduction via inverse regression: A minimum discrepancy approach

R. Dennis Cook
, Liqiang Ni

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

267 Scopus citations

Abstract

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 language	English (US)
Pages (from-to)	410-428
Number of pages	19
Journal	Journal of the American Statistical Association
Volume	100
Issue number	470
DOIs	https://doi.org/10.1198/016214504000001501
State	Published - 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.

Keywords

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

Publisher link

10.1198/016214504000001501

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title = "Sufficient dimension reduction via inverse regression: A minimum discrepancy approach",

abstract = "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.",

keywords = "Inverse regression estimator, Sliced average variance estimation, Sliced inverse regression, Sufficient dimension reduction",

author = "Cook, \{R. Dennis\} and Liqiang Ni",

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.",

year = "2005",

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doi = "10.1198/016214504000001501",

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PY - 2005/6

Y1 - 2005/6

N2 - 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.

AB - 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.

KW - Inverse regression estimator

KW - Sliced average variance estimation

KW - Sliced inverse regression

KW - Sufficient dimension reduction

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JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

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ER -

Sufficient dimension reduction via inverse regression: A minimum discrepancy approach

Abstract

Bibliographical note

Keywords

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