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Estimation and variable selection for generalized functional partially varying coefficient hybrid models

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Abstract

In this article, we propose a novel class of generalized functional partially varying coefficient hybrid models and variable selection procedure in which the explanatory variables include infinite dimensional predictor processes, treated as functional data with measurement errors, and high-dimensional scalar covariates with a diverging number of parameters. We focus on estimating coefficients and selecting the important variables in the high-dimensional covariates, which is complicated by the infinite-dimensional functional predictor, and one of our contributions is to characterize the effects of regularization on the resulting estimators. The proposed method is based on B-spline basis and functional principal component basis function approximation and a class of variable selection procedures using nonconcave penalized likelihood. Under some regularity conditions, we establish the consistency and oracle properties of the resulting shrinkage estimator, and empirical illustrations are given by simulation and illustrate its application using the biscuit dough dataset.

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References

  • Aneiros PG, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76:1102–1110

    Article  MathSciNet  Google Scholar 

  • Breheny P, Huang J (2015) Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors. Stat Comput 25:173–187

    Article  MathSciNet  Google Scholar 

  • Brown PJ, Fearn T, Vannucci M (2001) Bayesian wavelet regression on curves with application to a spectroscopic calibration problem. J Am Stat Assoc 96:398–408

    Article  MathSciNet  Google Scholar 

  • Cai TT, Yuan M (2012) Minimax and adaptive prediction for functional linear regression. J Am Stat Assoc 107:1201–1216

    Article  MathSciNet  Google Scholar 

  • Cao RY, Du J, Zhou JJ, Xie TF (2020) FPCA-based estimation for generalized functional partially linear models. Stat Pap 61:2715–2735

    Article  MathSciNet  Google Scholar 

  • Cardot H, Sarda P (2008) Varying-coefficient functional linear regression models. Commun Stat Theory Methods 37:3186–3203

    Article  MathSciNet  Google Scholar 

  • Cardot H, Ferraty F, Mas A, Sarda P (2003) Testing hypotheses in the functional linear model. Scand J Stat 30:241–255

    Article  MathSciNet  Google Scholar 

  • Carroll RJ, Fan J, Gijbels I, Wand MP (1997) Generalized partially linear single-index models. J Am Stat Assoc 92:477–489

    Article  MathSciNet  Google Scholar 

  • Cuevas A, Febrero M, Fraiman R (2002) Linear functional regression: the case of fixed design and functional response. Can J Stat 30:285–300

    Article  MathSciNet  Google Scholar 

  • Ding H, Zhang RQ, Zhang J (2018) Quantile estimation for a hybrid model of functional and varying coefficient regressions. J Stat Plan Inference 196:1–18

    Article  MathSciNet  Google Scholar 

  • Du J, Xu DK, Cao RY (2018) Estimation and variable selection for partially functional linear models. J Korean Stat Soc 47:436–449

    Article  MathSciNet  Google Scholar 

  • Fan JQ, Li RZ (2001) Variable selection via nonconcave penalized likelihood and its Oracle properties. J Am Stat Assoc 96:1348–1360

    Article  MathSciNet  Google Scholar 

  • Fan JQ, Li RZ (2004) New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. J Am Stat Assoc 99:710–723

    Article  MathSciNet  Google Scholar 

  • Feng SY, Xue LG (2016) Partially functional linear varying coefficient model. Statistics 50:717–732

    Article  MathSciNet  Google Scholar 

  • Fox J (2015) Applied regression analysis and generalized linear models. SAGE Publications, Inc., Thousand Oaks

    Google Scholar 

  • Guo CH, Yang H, Lv J (2017) Robust variable selection in high-dimensional varying coefficient models based on weighted composite quantile regression. Stat Pap 58:1009–1033

    Article  MathSciNet  Google Scholar 

  • Hall P, Horowitz JL (2007) Methodology and convergence rates for functional linear regression. Ann Stat 35:70–91

    Article  MathSciNet  Google Scholar 

  • Huang J, Horowitz JL, Wei FR (2010) Variable selection in nonparametric additive models. Ann Stat 38:2282–2313

    Article  MathSciNet  Google Scholar 

  • Kai B, Li RZ, Zou H (2011) New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann Stat 39:305–332

    Article  MathSciNet  Google Scholar 

  • Kong DH, Xue KJ, Yao F, Zhang HH (2016) Partially functional linear regression in high dimensions. Biometrika 103:147–159

    Article  MathSciNet  Google Scholar 

  • Ma HQ, Li T, Zhu HT, Zhu ZY (2019) Quantile regression for functional partially linear model in ultra-high dimensions. Comput Stat Data Anal 129:135–147

    Article  MathSciNet  Google Scholar 

  • Morris Jeffrey S (2015) Functional regression. Annu Rev Stat Appl 2:321–359

    Article  Google Scholar 

  • Pollard D (1991) Asymptotics for least absolute deviation regression estimators. Econom Theory 7:186–199

    Article  MathSciNet  Google Scholar 

  • Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New York

    Book  Google Scholar 

  • Sherwood B, Wang L (2016) Partially linear additive quantile regression in ultra-high dimension. Ann Stat 44:288–317

    Article  MathSciNet  Google Scholar 

  • Shin H (2009) Partial functional linear regression. J Stat Plan Inference 139:3405–3418

    Article  MathSciNet  Google Scholar 

  • van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Wang HS, Xia YC (2009) Shrinkage estimation of the varying coefficient model. J Am Stat Assoc 104:747–757

    Article  MathSciNet  Google Scholar 

  • Wang L, Liu X, Liang H, Carroll RJ (2011) Estimation and variable selection for generalized additive partial linear models. Ann Stat 39:1827–1851

    Article  MathSciNet  Google Scholar 

  • Wang JL, Chiou JM, Müller HG (2016) Functional data analysis. Annu Rev Stat Appl 3:257–295

    Article  Google Scholar 

  • Wei F, Huang J, Li HZ (2011) Variable selection and estimation in high-dimensional varying-coefficient models. Stat Sin 21:1515–1540

    Article  MathSciNet  Google Scholar 

  • Yao F, Müller HG, Wang JL (2005) Functional linear regression analysis for longitudinal data. Ann Stat 33:2873–2903

    Article  MathSciNet  Google Scholar 

  • Zhao PX, Xue LG (2009) Variable selection for semiparametric varying coefficient partially linear models. Stat Probab Lett 79:2148–2157

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work was partially supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (22XNL016).

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Authors and Affiliations

  1. College of Science, North China Institute of Science and Technology, Sanhe, 065201, China

    Yanxia Liu

  2. Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing, 100872, China

    Yanxia Liu, Zhihao Wang & Maozai Tian

  3. School of Statistics and Data Science, Xinjiang University of Finance and Economics, Ürümqi, 830012, China

    Zhihao Wang & Maozai Tian

  4. Department of Medical Engineering and Technology, Xinjiang Medical University, Ürümqi, 830011, China

    Maozai Tian

  5. School of Statistics, Lanzhou University of Finance and Economics, Lanzhou, 730010, China

    Maozai Tian

  6. Mathematical Sciences, Brunel University, Uxbridge, UB83PH, UK

    Keming Yu

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  1. Yanxia Liu
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  2. Zhihao Wang
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  3. Maozai Tian
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  4. Keming Yu
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Correspondence to Maozai Tian.

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Liu, Y., Wang, Z., Tian, M. et al. Estimation and variable selection for generalized functional partially varying coefficient hybrid models. Stat Papers 65, 93–119 (2024). https://doi.org/10.1007/s00362-022-01383-z

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