Identification and estimation for generalized varying coefficient partially linear models

Year 2018, Volume: 47 Issue: 4, 1041 - 1060, 01.08.2018

Mingqiu Wang Xiuli Wang Muhammad Amin

Abstract

The generalized varying coefficient partially linear model (GVCPLM) enjoys the flexibility of the generalized varying coefficient model and the parsimony and interpretability of the generalized linear model. Statistical inference of GVCPLM is restricted with a condition that the components of varying and constant coefficients are known in advance. Alternatively, the current study is focused on the structure's identification of varying and constant coefficient for GVCPLM and it is based on the spline basis approximation and the group SCAD. This is proved that the proposed method can consistently determine the structure of the GVCPLM under certain conditions, which means that it can accurately choose the varying and constant coefficients precisely. Simulation studies and a real data application are conducted to assess the infinite sample performance of the proposed method.

Keywords

Group variable selection, Group SCAD, Selection consistency, Structure identification

References

• Ahmad, I, Leelahanon, S, Li, Q. Efficient estimation of a semiparametric partially linear varying coecient model. Annals of Statistics. 33, 258283, 2005.
• Cai, Z, Fan, J, Li, R. Efficient estimation and inferences for varying-coefficient models. Journal of the American Statistical Association. 95, 888902, 2000.
• Fan, J, Li, R. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association. 6, 13481360, 2001.
• Hastie, T, Tibshirani, R. Varying-coefficient models. Journal of the Royal Statistical Society, Series B. 55, 757796, 1993.
• Hu, T, Cui, H. Robust estimates in generalised varying-coefficient partially linear models. Journal of Nonparametric Statistics. 22, 737754, 2010.
• Hu, T, Xia, Y. Adaptive semi-varying coefficient model selection. Statistica Sinica. 22, 575 599, 2012.
• Huang, J, Horowitz, JL, Wei FR. Variable selection in nonparametric additive models. Annals of Statistics. 38, 22822313, 2010.
• Huang, J, Wei, F, Ma, S. Semiparametric regression pursuit. Statistica Sinica. 22, 1403 1426, 2012.
• Huang, JHZ, Wu, CO, Zhou, L. Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statistica Sinica. 14, 763788, 2004.
• Kim, Y, Choi, H, Oh, H. Smoothly clipped absolute deviation on high dimensions. Journal of the American Statistical Association. 103, 16561673, 2008.
• Kai, B, Li, R, Zou, H. New efficient estimation and variable methods for semiparametric varying-coefficient partially linear models. Annals of Statistics. 39, 305332, 2011.
• Lam, C, Fan, J. Prole-kernel likelihood inference with diverging number of parameters. Annals of Statistics. 36, 22322260, 2008.
• Li, Q, Huang, CJ, Li, D, Fu, TT. Semiparametric smooth coefficient models. Journal of Business and Economic Statistics. 20, 412422, 2002.
• Li, R, Liang, H. Variable selection in semiparametric regression modeling. Annals of Statis- tics. 36, 261286, 2008.
• Li G, Xue L, Lian H. SCAD-penalised generalised additive models with non-polynomial dimensionality. Journal of Nonparametric Statistics, 24, 681697, 2012.
• Li, G, Lin, L, Zhu, L. Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters. Journal of Multivariate Analysis. 105, 85111, 2012.
• Lian, H. Variable selection for high-dimensional generalized varying-coefficient models. Sta- tistica Sinica. 22, 15631588, 2012.
• Lian, H, Chen, X, Yang, JY. Identication of partially linear structure in additive models with an application to gene expression prediction from sequences. Biometrics. 68, 437445, 2012.
• Lian, H, Du, P, Li, Y, Liang, H. Partially linear structure identication in generalized additive models with NP-dimensionality. Computational Statistics & Data Analysis. 80, 197208, 2014.
• Lian, H, Liang, H, Ruppert, D. Separation of covariates into nonparametric and parametric parts in high-dimensional partially linear additive models. Statistica Sinica. 25, 591607, 2015.
• Lu, Y:. Generalized partially linear varying-coefficient models. Journal of Statistical Plan- ning and Inference. 138, 901914, 2008.
• McCullagh, P, Nelder, JA. Generalized Linear Models. Chapman and Hall, London (1989).
• Schwarz, G. Estimating the dimension of a model. Annals of Statistics. 6, 461464, 1978.
• Schumaker, LL. Spline Functions: Basic Theory. Wiley, New York (1981).
• Tang, Y, Wang, HJ, Zhu, Z, Song, X. A unified variable selection approach for varying coefficient models. Statistica Sinica. 22, 601628, 2012.
• Wang, D, Kulasekera, KB. Parametric component detection and variable selection in varying-coefficient partially linear models. Journal of Multivariate Analysis. 112, 117129, 2012.
• Wang, H., Li, R., Tsai, C.L. Tuning Parameter Selectors for the Smoothly Clipped Absolute Deviation Method. Biometrika, 94, 553568, 2007.
• Wang, M, Song, L. Identication for semiparametric varying coefficient partially linear models. Statist. Statistics & Probability Letters. 83, 13111320, 2013.
• Xia, Y, Zhang, W, Tong, H. Efficient estimation for semivarying-coefficient models. Biometrika. 91, 661681, 2004.
• Zhang, H, Cheng, G, Liu, Y. Linear or nonlinear? Automatic structure discovery for par- tially linear models. Journal of the American Statistical Association. 106, 10991112, 2011.
• Zhou ,S, Shen, X,Wolfe, DA. Local asymptotics for regression splines and confidence regions. Annals of Statistics. 26, 17601782, 1998.