Sparse dimension reduction with group identification

Document Type : Research Paper


Department of Statistics, College of Administration and Economics, University of Al-Qadisiyah, Al Diwaniyah, Iraq.


 Estimating the central mean subspace without requiring to designate a model is achieved via MAVE method. The original $p$ predictors are replaced with $d$-linear combinations (LC) of predictors in MAVE, where $d<p$  without loss of any information about the regression. However, it is known that the interpretation of the estimated effective dimension reduction (EDR) direction is not easy due to each EDR direction is a LC of all the original predictors. The PACS method is an oracle procedure. In this article, a group variable selection method (SMAVE-PACS) is proposed. The sufficient dimension reduction (SDR) concepts and group variable selection are emerged through SMAVE-PACS. SMAVE-PACS produces sparse and accurate solutions with the ability of group identification. SMAVE-PACS extended PACS to multi-dimensional regression under SDR conditions. In addition, a method for estimating the structural dimension was proposed. The effectiveness of the SMAVE-PACS is checked through simulation and real data.


[1] A. Alkenani and K. Yu, Sparse MAVE with oracle penalties, Adv. Appl. Stat., 34(2013) 85-105.
[2] R. Cook, Regression graphics: ideas for studying the regression through graphics, New York, Wily, 1998.
[3] R. D. Cook and B. Li Dimension reduction for the conditional mean in regression, Ann. Stat., 30(2002) 455-474.
[4] R. D. Cook, and S. Weisberg, Discussion of Li , J. Am. Stat Assoc., 86 (1991) 328-332.
[5] J. Fan and R. Z.Li, Variable selection via non-concave penalized likelihood and its oracle properties, J. Am. Stat.
Assoc., 96 (2001) 1348-1360.
[6] L. Li, Sparse sufficient dimension reduction, Biometrika, 94 (2007) 603-613.
[7] K. Li, Sliced inverse regression for dimension reduction (with discussion), J. Am. Stat. Assoc., 86(1991) 316-342.
[8] K. C.Li, On principal Hessian directions for data visualization and dimension reduction: Another application of
Stein’s lemma, J. Am. Stat. Assoc., 87(1992) 1025-1039.
[9] L. Li, R. D.Cook and C. J.Nachtsheim, Model-free variable selection, J. R. Stat. Soc. Ser. B, 67(2005) 285-299.
[10] L.Li and C. J. Nachtsheim, Sparse sliced inverse regression, Technometrics , 48(2006) 503-510.
[11] L. Li and X. Yin, Sliced Inverse Regression with regularizations, Biometrics, 64(2008) 124-131.
[12] W. D. Mangold, L. Bean, and D. Adams, The impact of intercollegiate athletics on graduation rates among
major NCAA Division I universities: Implications for college persistence theory and practice, J. Higher Educ,
74(5)(2003) 540-562.
[13] G. C. McDonald and R. C. Schwing, Instabilities of regression estimates relating air pollution to mortality,
Technometrics, 15(3) (1973) 463-481.
[14] L. Ni, R. D. Cook and C. L. Tsai, A note on shrinkage sliced inverse regression ,Biometrika , 92(2005) 242-247.
[15] D. B. Sharma, H. D. Bondell and H. H.Zhang, Consistent group identification and variable selection in regression
with correlated predictors, J. Comput. Graphical Stat., 22(2)(2013) 319-340.
[16] B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, 1986.
[17] R. Tibshirani, Regression shrinkage and selection via the Lasso, J. Royal Stat. Soc. Ser. B, 58 (1996) 267-288.
[18] Q. Wang and X. Yin, A Nonlinear Multi-Dimensional Variable Selection Method for High Dimensional Data:
Sparse MAVE, Comput. Stat. Data Anal. , 52(2008) 4512-4520.
[19] T. Wang, P. Xu and L. Zhu, Variable selection and estimation for semiparametric multiple-index models, Bernoulli,
21( 1) (2015) 242-275.
[20] Y. Xia, H. Tong, W. Li and L.Zhu, An adaptive estimation of dimension reduction space, J. Royal Stat. Soc. Ser.
B , 64(2002)363-410.
[21] Z. Yu and L. Zhu, Dimension reduction and predictor selection in semiparametric models, Biometrika, 100 (2013)
[22] T. Wang, P. Xu and L. Zhu, Penalized minimum average variance estimation, Statist. Sinica , 23(2013) 543-569.
[23] C. H. Zhang, Nearly unbiased variable selection under minimax concave penalty, Annal. Stat., 38 (2010) 894-942.
[24] H. Zou, The adaptive Lasso and its oracle properties. J. Am. Stat. Assoc., 101(2006) 1418-142.
[25] H. Zou and T. Hastie, Regularization and variable selection via the elastic net, J. Royal Stat. Soc., Ser. B ,67(2005)
Volume 13, Issue 1
March 2022
Pages 2921-2931
  • Receive Date: 21 May 2021
  • Revise Date: 05 October 2021
  • Accept Date: 18 October 2021