Robust estimation and variable selection in sufficient dimension reduction. (English) Zbl 1466.62182

Summary: Dimension reduction and variable selection play important roles in high dimensional data analysis. Minimum Average Variance Estimation (MAVE) is an efficient approach among many others. However, because of the use of least squares criterion, MAVE is not robust to outliers in the dependent variable or errors with heavy tailed distributions. A robust extension of MAVE through modal regression is proposed. This new approach can adapt to different error distributions and thus brings robustness to the contamination in the response variable. The estimator is shown to have the same convergence rate as the original MAVE. Furthermore, the proposed method is combined with adaptive LASSO to select informative variables. The efficacy of this new solution is illustrated through simulation studies and a data analysis on Hong Kong air quality.


62-08 Computational methods for problems pertaining to statistics


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[1] Čížek, P.; Härdle, W., Robust estimation of dimension reduction space, Comput. Stat. Data Anal., 51, 545-555, (2006) · Zbl 1157.62382
[2] Cook, R. D., Regression graphics: ideas for studying regressions through graphics, (1998), Wiley New York · Zbl 0903.62001
[3] Cook, R. D.; Li, B., Dimension reduction for the conditional mean in regression, Ann. Statist., 30, 455-474, (2002) · Zbl 1012.62035
[4] Cook, R. D.; Weisberg, S., Comment, J. Amer. Statist. Assoc., 86, 328-332, (1991) · Zbl 1353.62037
[5] Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, with applications, J. Roy. Statist. Soc. Ser. B, 39, 1-38, (1977)
[6] Fan, J.; Jiang, J., Variable bandwidth and one-step local M-estimator, Sci. China Ser. A, 43, 65-81, (2000) · Zbl 0969.62028
[7] Friedman, J. H.; Hastie, T.; Tibshirani, R., Regularization paths for generalized linear models via coordinate descent, J. Stat. Softw., 33, 1, 1-22, (2010)
[8] Härdle, W.; Stoker, T. M., Investigating smooth multiple regression by the method of average derivatives, J. Amer. Statist. Assoc., 84, 986-995, (1989) · Zbl 0703.62052
[9] Hristache, M.; Juditsky, A.; Polzehl, J.; Spokoiny, V., Structure adaptive approach for dimension reduction, Ann. Statist., 29, 1537-1566, (2001) · Zbl 1043.62052
[10] Li, K. C., Sliced inverse regression for dimension reduction, J. Amer. Statist. Assoc., 86, 316-327, (1991) · Zbl 0742.62044
[11] Li, K. C., On principal Hessian directions for data visualization and dimension reduction: another application of stein’s lemma, J. Amer. Statist. Assoc., 87, 1025-1039, (1992) · Zbl 0765.62003
[12] Li, K. C.; Duan, N. H., Regression analysis under link violation, Ann. Statist., 17, 1009-1052, (1989) · Zbl 0753.62041
[13] Li, B.; Zha, H.; Chiaromonte, F., Contour regression: a general approach to dimension reduction, Ann. Statist., 1580-1616, (2005) · Zbl 1078.62033
[14] Silverman, B., Density estimation for statistics and data analysis, (1986), Chapman & Hall London · Zbl 0617.62042
[15] Wang, H.; Xia, Y., Sliced regression for dimension reduction, J. Amer. Statist. Assoc., 103, 811-821, (2008) · Zbl 1306.62168
[16] Wang, Q.; Yao, W., An adaptive estimation of MAVE, J. Multivariate Anal., 104, 88-100, (2012) · Zbl 1352.62060
[17] Wang, Q.; Yin, X., A nonlinear multi-dimensional variable selection method for high dimensional data: sparse MAVE, Comput. Statist. Data Anal., 52, 4512-4520, (2008) · Zbl 1452.62136
[18] Welsh, A. H.; Ronchetti, E., A journey in single steps: robust one-step M-estimation in linear regression, J. Statist. Plann. Inference, 103, 287-310, (2002) · Zbl 0988.62040
[19] Xia, Y.; Tong, H.; Li, W. K.; Zhu, L. X., An adaptive estimation of dimension reduction space, J. R. Stat. Soc. Ser. B Stat. Methodol., 64, 363-410, (2002) · Zbl 1091.62028
[20] Yao, W.; Lindsay, B. G.; Li, R., Local modal regression, J. Nonparametr. Stat., 24, 647-663, (2012) · Zbl 1254.62059
[21] Yao, W.; Wang, Q., Robust variable selection through MAVE, Comput. Statist. Data Anal., 63, 42-49, (2013)
[22] Ye, Z.; Weiss, R. E., Using the bootstrap to select one of a new class of dimension reduction methods, J. Amer. Statist. Assoc., 98, 968-979, (2003) · Zbl 1045.62034
[23] Yin, X.; Li, B.; Cook, R. D., Successive direction extraction for estimating the central subspace in a multiple-index regression, J. Multivariate Anal., 99, 1733-1757, (2008) · Zbl 1144.62030
[24] Zou, H., The adaptive lasso and its oracle properties, J. Amer. Statist. Assoc., 101, 1418-1429, (2006) · Zbl 1171.62326
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