Abstract

We study generalized functional partially additive hybrid model (GF-

PAHM) where the explanatory variables involve both infinite dimensional predictor

processes viewed as functional data with measurement errors, and high-dimensional

scalar covariates whose impact on the response is nonlinear. Despite extensive work

focusing on functional linear models, little effort has been devoted to estimate coefficients and selecting the important additive components for the GFPAHM, which

is complicated by the infinite-dimensional functional predictor. We investigate a

nonconvex penalized likelihood estimator for simultaneous variable selection and estimation. The proposed method and theoretical development are quite challenging

since the numbers of nonlinear components increase as the sample size increases.

Asymptotic properties of the proposed shrinkage estimators are investigated. Extensive Monte Carlo simulations have been conducted and show that the proposed

procedure works effectively even with moderate sample sizes, and analyze the biscuit dough data set as an illustration.

Information

Preprint No.SS-2024-0358
Manuscript IDSS-2024-0358
Complete AuthorsYanxia Liu, Zhihao Wang, Yu Zhen, Wolfgang K. Härdle, Maozai Tian
Corresponding AuthorsMaozai Tian
Emailsmztian@ruc.edu.cn

References

  1. Aneiros, P. G. and P. Vieu (2006). Semi-functional partial linear regression. Statistics and Probability Letters 76, 1102–1110.
  2. Breheny, P. and J. Huang (2015). Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors. Statistics and Computing 25, 173–187.
  3. Brown, P. J., T. Fearn, and M. Vannucci (2001). Bayesian wavelet regression on curves with application to a spectroscopic calibration problem. Journal of the American Statistical Association 96, 398–408.
  4. Cao, R. Y., J. Du, J. J. Zhou, and T. F. Xie (2020). FPCA-based estimation for generalized functional partially linear models. Statistical Papers 61, 2715–2735.
  5. Cardot, H., F. Ferraty, A. Mas, and P. Sarda (2003). Testing hypotheses in the functional linear model. Scandinavian Journal of Statistics 30, 241–255.
  6. Cardot, H., F. Ferraty, P. Sarda, and Toulouse. (2003). Spline estimators for the functional linear model. Statistica Sinica 13, 571–591.
  7. Cardot, H. and P. Sarda (2008). Varying-coefficient functional linear regression models. Communications in Statistics-Theory and Methods 37, 3186–3203.
  8. Carroll, R. J.and Fan, J. Q., I. Gijbels, and M. P. Wand (1997). Generalized partially linea single-index models. Journal of the American Statistical Association 92, 477–489.
  9. Cuevas, A., M. Febrero, and R. Fraiman (2002). Linear functional regression: The case of fixed design and functional response. The Canadian Journal of Statistics 30, 285–300.
  10. de Boor, C. (2001). A Practical Guide to Splines. New York: Springer.
  11. Delaigle, A. and P. Hall (2012). Methodology and theory for partial least squares applied to functional data. The Annals of Statistics 40, 322–352.
  12. Ding, H., R. Q. Zhang, and J. Zhang (2018). Quantile estimation for a hybrid model of functional and varying coefficient regressions. Journal of Statistical Planning and Inference 196, 1–18.
  13. Du, J., R. Y. Cao, E. Kwessi, and Z. Z. Zhang (2019). Estimation for generalized partially functional linear additive regression model. Journal of Applied Statistics 46, 914–925.
  14. Du, J., D. K. Xu, and R. Y. Cao (2018). Estimation and variable selection for partially functional linear models. Journal of the Korean Statistical Society 47, 436–449.
  15. Fan, J. Q. and R. Z. Li (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360.
  16. Fan, J. Q. and R. Z. Li (2004). New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. Journal of the American Statistical Association 99, 710–723.
  17. Feng, S. Y. and L. G. Xue (2016). Partially functional linear varying coefficient model. Statistics 50, 717–732.
  18. Guo, C. H., H. Yang, and J. Lv (2017). Robust variable selection in high-dimensional varying coefficient models based on weighted composite quantile regression. Statistical Papers 58, 1009–1033.
  19. Hall, P. and J. L. Horowitz (2007). Methodology and convergence rates for functional linear regression. Annals of Statistics 35, 70–91.
  20. Hsing, T. and R. L. Eubank (2015). Theoretical Foundations of Functional Data Analysis with an Introduction to Linear Operators. Ltd: John Wiley and Sons.
  21. Huang, J., J. L. Horowitz, and F. R. Wei (2010). Variable selection in nonparametric additive models. Annals of Statistics 38, 2282–2313.
  22. Kai, B., R. Z. Li, and H. Zou (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Annals of Statistics 39, 305–332.
  23. Kong, D. H., K. J. Xue, F. Yao, and H. H. Zhang (2016). Partially functional linear regression in high dimensions. Biometrika 103, 147–159.
  24. Liu, Y. X., Z. H. Wang, M. Z. Tian, and K. Yu (2022). Estimation and variable selection for generalized functional partially varying coefficient hybrid models. Statistical Papers.
  25. Morris and S. Jeffrey (2015). Functional regression. Annual Review of Statistics and Its Application 2, 321–359.
  26. Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.
  27. Ramsay, J. O. (1982). When the data are functions. Psychometrika 47, 379–396.
  28. Ramsay, J. O. and B. W. Silverman (2005). Functional data analysis. New York: Springer.
  29. Reiss, P. T. and R. T. Ogden (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistical Association 102, 984–996.
  30. Severini, T. A. and J. G. Staniswalis (1994). Quasi-likelihood estimation in semiparametricmodels. Journal of the American Statistical Association 89, 501–511.
  31. Sherwood, B. and L. Wang (2016). Partially linear additive quantile regression in ultra-high dimension. Annals of Statistics 44, 288–317.
  32. Shin, H. (2009). Partial functional linear regression. Journal of Statistical Planning and Inference 139, 3405– 3418.
  33. Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Annals of Statistics 14, 590–606.
  34. Tang, Q. G., W. Tu, and L. L. Kong (2023). Estimation for partial functional partially linear additive model. Computational Statistics and Data Analysis 177, 1–15.
  35. Tibshirani, R. J. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B 58, 267–288.
  36. van Der Vaart, A. W. (1998). Asymptotic statistics.
  37. Wang, H. S. and Y. C. Xia (2009). Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association 104, 747–757.
  38. Wang, J. L., J. M. Chiou, and H. G. M¨uller (2016). Functional data analysis. Annual Review of Statistics and Its Application 3, 257–295.
  39. Wang, L., X. Liu, H. Liang, and R. J. Carroll (2011). Estimation and variable selection for generalized additive partial linear models. Annals of Statistics 39, 1827–1851.
  40. Wei, F., J. Huang, and H. Z. Li (2011). Variable selection and estimation in high-dimensional varying-coefficient models. Statistica Sinica 21, 1515–1540.
  41. Wong, R. K. W., Y. H. Li, and Z. Y. Zhu (2018). Partially linear functional additive models for multivariate functional data. Journal of the American Statistical Association 114, 406–418.
  42. Wu, Y. C., J. Q. Fan, and H. G. M¨uller (2010). Varying-coefficient functional linear regression. Bernoulli 16(3), 730–758.
  43. Yao, F., H. G. M¨uller, and J. L. Wang (2005). Functional linear regression analysis for longitudinal data. Annals of Statistics 33, 2873–2903.
  44. Yao, F., S. Sue Chee, and F. Wang (2017). Regularized partially functional quantile regression. Journal of Multivariate Analysis 159, 39–56.
  45. Zhang, D., X. Lin, and M. F. Sowers (2007). Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome. Biometrics 63, 351–362.
  46. Zhao, P. X. and L. G. Xue (2009). Variable selection for semiparametric varying coefficient partially linear models. Statistics and Probability Letters 79, 2148–2157. Yanxia Liu, College of Science, North China Institute of Science and Technology, Hebei, 065201, China

Acknowledgments

The work was partially supported by the Beijing Natural Science Foundation (No.1242005),

the Fundamental Research Funds for the Central Universities, and the Research Funds

of Renmin University of China (25XNN015), the Fundamental Research Funds for

the Central Universities of China(3142023039); the Science and Technology Support

Project of Langfang(2024011081, 2024011084); Key Discipline of Applied Statistics of

North China Institute of Science and Technology; Natural Science Foundation of Xinjiang Uygur Autonomous Region(2023D01A74); the Social Science Fund of Xinjiang

Uygur Autonomous Region (2023BTJ048); National Social Science Fund (22XTJ006);

High-level Talent Special Project of Xinjiang University of Finance and Economics

Professor Wolfgang H¨ardle’s work is supported through the European Cooperation

in Science & Technology COST Action grant CA19130 - Fintech and Artificial Intelligence in Finance - Towards a transparent financial industry; the project “IDA Institute

of Digital Assets”, CF166/15.11.2022, contract number CN760046/23.05.2024 financed

under the Romania’s National Recovery and Resilience Plan, Apel nr. PNRR-III-C9-

2022-I8; and the Marie Sklodowska-Curie Actions under the European Union’s Horizon

Europe research and innovation program for the Industrial Doctoral Network on Digital

Finance, acronym DIGITAL, Project No. 101119635.; the Yushan Fellowships, TW.

Supplementary Materials

The online supplementary material contains the technical proofs for the theorems for

the proposed methodology.

Supplementary materials are available for download.