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Volume 6, Issue 1, March 2020, Page: 1-9
Variable Selection for Partially Linear Additive Model Based on Modal Regression Under High Dimensional Data
Yafeng Xia, School of Sciences, Lanzhou University of Technology, Lanzhou, P. R. China
Lirong Zhang, School of Sciences, Lanzhou University of Technology, Lanzhou, P. R. China
Received: Dec. 19, 2019;       Accepted: Jan. 9, 2020;       Published: Apr. 17, 2020
Abstract
In this article, we focus on the variable selection for partially linear additive model under high dimensional data. Variable selection is proposed based on modal regression estimation with Adoptive Bridge Method. Using the B-spline basic function to approximate the additive function, a penalty estimation objective equation is constructed. It establishes and proves that the variable selection methods have oracle property. Numerical simulations tested the performance of the proposed methods in a finite sample and verified the significance of the proposed estimation and the variable selection methods. At the end of the article, we attach the detailed derivation of the theoretical results. Therefore, the correctness of the method used is verified theoretically and practically.
Keywords
High Dimensional Data, Partially Linear Additive Model, Modal Regression, Variable Selection, Adoptive Bridge, B-spline
Yafeng Xia, Lirong Zhang, Variable Selection for Partially Linear Additive Model Based on Modal Regression Under High Dimensional Data, International Journal of Statistical Distributions and Applications. Vol. 6, No. 1, 2020, pp. 1-9. doi: 10.11648/j.ijsd.20200601.11
Reference
[1]
Hastie, T. J, Tibshirani, R. J. (2017). Generalized Additive Models. New York, Routledge. doi: 10.1201/9780203753781.
[2]
Guo, J, Tang, M. L, Tian, M. Z, Zhu, K. (2013).Variable selection in high-dimensional partially linear additive models for composite quantile regression. Computational Statistics and Data Analysis, 65 (9), 56. doi: 10.1016/j.csda.2013.03.017.
[3]
Wang, L, Liu, X, Liang, H, Raymond, J. C. (2011). Estimation and variable selection for semiparmetric additive partial linear models. Statistica Sinica, 21 (3), 1225. doi: 10.2307/23033585.
[4]
Xia, Y. F, Qu, Y. R, Sun, N. L. (2018). Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data. Communications in Statistics Theory and Methods, 48 (20), 5121. doi: 10.1080/03610926.2018.1508712.
[5]
Fan, J. Q, Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 11 (6), 1031. doi: 10.2307/25464778.
[6]
Hoshino, T. (2014). Quantile regression estimation of partially linear additive models. Journal of Non-parametric Statistics, 26 (3), 509. \\doi: 10.1080/10485252.2014.929675.
[7]
Meinshausen, N, Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. The Annals of statistics, 34 (3), 1436. doi: 10.1214/009053606000000281.
[8]
Zhang, C. H, Huang, J. (2008). The sparsity and bias of the lasso selection in high-dimensional linear regression. The Annals of Statistics, 36 (4), 1567. doi: 10.1214/07-AOS520.
[9]
Wang, M. Q, Song, L. X, Wang, X. G. (2010). Bridge estimation for generalized linear models with a diverging number of parameters. Statistics Probability Letters, 80 (21), 1584. doi: 10.1016/j.spl.2010.06.012.
[10]
Li, K. P, Li, D. G, Liang, Z. W. (2017). Estimation of semi-varying coefficient models with nonstationary regressors. Econometric Reviews, 36 (1), 354. doi: 10.1080/07474938.2015.1114563.
[11]
Lam, C, Fan, J. Q. (2008). Profile-kernel likelihood inference with diverging number of parameters. The Annals of Statistics, 36 (5), 2232. doi: 10.1214/07-AOS544.
[12]
Yao, W. X, Lindsay, B. G, Li, R. Z. (2012). Local modal regression. Journal of Nonparametric Statistics, 24 (3), 647. doi: 10.1080/10485252.2012.678848.
[13]
Fan, J. Q, Li, R. Z. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association, 96 (456), 1348. doi: 10.1198/016214501753382273.
[14]
Zhao, P. X, Xue, L. G. (2009). Variable selection for semiparametric varying coefficient partially linear models. Statistics Probability Letters, 79 (20), 2148. doi: 10.1016/j.spl.2009.07.004.
[15]
Schumaker, L. L. (2007).Splines Function: Basic Theory. Cambridge, Cambridge University Press. doi: 10.1017/CBO9780511618994.