TY - JOUR
T1 - Dimension reduction in regressions through cumulative slicing estimation
AU - Zhu, Li-Ping
AU - Zhu, Li-Xing
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: [email protected]). 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
UR - http://www.scopus.com/inward/record.url?scp=78651340680&partnerID=8YFLogxK
U2 - 10.1198/jasa.2010.tm09666
DO - 10.1198/jasa.2010.tm09666
M3 - Journal article
AN - SCOPUS:78651340680
SN - 0162-1459
VL - 105
SP - 1455
EP - 1466
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 492
ER -