Dimension reduction in regressions through cumulative slicing estimation

Li Ping Zhu; Lixing ZHU; Zheng Hui Feng

doi:10.1198/jasa.2010.tm09666

Dimension reduction in regressions through cumulative slicing estimation

Li Ping Zhu^*, Lixing ZHU, Zheng Hui Feng

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

119 Citations (Scopus)

Abstract

In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p = O(n^1/2/log n) and the asymptotic normality for p = o(n^1/2) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p = o(n^1/3) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p = o(n^1/2). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.

Original language	English
Pages (from-to)	1455-1466
Number of pages	12
Journal	Journal of the American Statistical Association
Volume	105
Issue number	492
DOIs	https://doi.org/10.1198/jasa.2010.tm09666
Publication status	Published - Dec 2010

Scopus Subject Areas

Statistics and Probability
Statistics, Probability and Uncertainty

User-Defined Keywords

Inverse regression
Slicing estimation
Sufficient dimension reduction
Ultrahigh dimensionality

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10.1198/jasa.2010.tm09666

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title = "Dimension reduction in regressions through cumulative slicing estimation",

abstract = "In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p = O(n1/2/log n) and the asymptotic normality for p = o(n1/2) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p = o(n1/3) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p = o(n1/2). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.",

keywords = "Inverse regression, Slicing estimation, Sufficient dimension reduction, Ultrahigh dimensionality",

author = "Zhu, {Li Ping} and Lixing ZHU and Feng, {Zheng Hui}",

note = "Funding Information: Liping Zhu is currently a Faculty Member of School of Statistics and Management, Shanghai University of Finance and Economics. He is supported by a NSF grant from National Natural Science Foundation of China (No. 10701035), SUFE through project 211 phase III and Shanghai Leading Academic Discipline Project (B 803). The first draft of this article was done when Liping Zhu was working at School of Finance and Statistics, East China Normal University. Lixing Zhu is Chair Professor, Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China (E-mail: lzhu@hkbu.edu.hk). His research was supported by a grant (HKBU2034/09P) from the Research Grants Council of Hong Kong and a FRG grant from Hong Kong Baptist University, Hong Kong. Zhenghui, Feng is Ph.D. Student, Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China. We are grateful to the Editor, an Associate Editor, and two referees for their helpful and constructive comments that substantially improved an earlier draft.",

year = "2010",

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doi = "10.1198/jasa.2010.tm09666",

language = "English",

volume = "105",

pages = "1455--1466",

journal = "Journal of the American Statistical Association",

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AU - Zhu, Li Ping

AU - ZHU, Lixing

AU - Feng, Zheng Hui

N1 - Funding Information: Liping Zhu is currently a Faculty Member of School of Statistics and Management, Shanghai University of Finance and Economics. He is supported by a NSF grant from National Natural Science Foundation of China (No. 10701035), SUFE through project 211 phase III and Shanghai Leading Academic Discipline Project (B 803). The first draft of this article was done when Liping Zhu was working at School of Finance and Statistics, East China Normal University. Lixing Zhu is Chair Professor, Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China (E-mail: lzhu@hkbu.edu.hk). His research was supported by a grant (HKBU2034/09P) from the Research Grants Council of Hong Kong and a FRG grant from Hong Kong Baptist University, Hong Kong. Zhenghui, Feng is Ph.D. Student, Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China. We are grateful to the Editor, an Associate Editor, and two referees for their helpful and constructive comments that substantially improved an earlier draft.

PY - 2010/12

Y1 - 2010/12

N2 - In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p = O(n1/2/log n) and the asymptotic normality for p = o(n1/2) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p = o(n1/3) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p = o(n1/2). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.

AB - In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p = O(n1/2/log n) and the asymptotic normality for p = o(n1/2) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p = o(n1/3) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p = o(n1/2). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.

KW - Inverse regression

KW - Slicing estimation

KW - Sufficient dimension reduction

KW - Ultrahigh dimensionality

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VL - 105

SP - 1455

EP - 1466

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 492

ER -

Dimension reduction in regressions through cumulative slicing estimation

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