Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Efficient estimation and variable selection for partially linear single-index-coefficient regression models

  • Kim, Young-Ju (Department of Statistics, Kangwon National University)
  • Received : 2018.11.20
  • Accepted : 2018.12.26
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A structured model with both single-index and varying coefficients is a powerful tool in modeling high dimensional data. It has been widely used because the single-index can overcome the curse of dimensionality and varying coefficients can allow nonlinear interaction effects in the model. For high dimensional index vectors, variable selection becomes an important question in the model building process. In this paper, we propose an efficient estimation and a variable selection method based on a smoothing spline approach in a partially linear single-index-coefficient regression model. We also propose an efficient algorithm for simultaneously estimating the coefficient functions in a data-adaptive lower-dimensional approximation space and selecting significant variables in the index with the adaptive LASSO penalty. The empirical performance of the proposed method is illustrated with simulated and real data examples.

Keywords

References

  1. Fan J and Li R (2001). Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association, 96, 1348-1360. https://doi.org/10.1198/016214501753382273
  2. Feng S and Xue L (2015). Model detection and estimation for single-index varying coefficient model, Journal of Multivariate Analysis, 139, 227-244. https://doi.org/10.1016/j.jmva.2015.03.008
  3. Foster JC, Taylor JMG, and Nan B (2013). Variable selection in monotone single-index models via the adaptive LASSO, Statistical Medicine, 32, 3944-3954. https://doi.org/10.1002/sim.5834
  4. Gu C (2013). Smoothing Spline ANOVA Models (2nd ed), Springer-Verlag.
  5. Gu C and Kim YJ (2002). Penalized likelihood regression: General formulation and efficient approximation, Canadian Journal of Statistics, 30, 619-628. https://doi.org/10.2307/3316100
  6. Huang Z (2012). Efficient inferences on the varying-coefficient single-index model with empirical likelihood, Computational Statistics and Data Analysis, 56, 4413-4420. https://doi.org/10.1016/j.csda.2012.03.024
  7. Huang Z, Lin B, Feng F, and Pang Z (2013). Efficient penalized estimating method in the partially varying-coefficient single-index model, Journal of Multivariate Analysis, 114, 189-200. https://doi.org/10.1016/j.jmva.2012.07.011
  8. Huang Z, Pang Z, Lin B, and Shao Q (2014). Model structure selection in single-index-coefficient regression models, Journal of Multivariate Analysis, 125, 159-175. https://doi.org/10.1016/j.jmva.2013.12.006
  9. Kim YJ and Gu C (2004). Smoothing spline Gaussian regression: more scalable computation via efficient approximation, Journal of the Royal Statistical Society Series B, 66, 337-356. https://doi.org/10.1046/j.1369-7412.2003.05316.x
  10. Leng C (2009). A simple approach for varying-coefficient model selection, Journal of Statistical Planning and Inference, 139, 2138-2146. https://doi.org/10.1016/j.jspi.2008.10.009
  11. Peng H and Huang T (2011). Penalized least squares for single index models, Journal of Statistical Planning and Inference, 141, 1362-1379. https://doi.org/10.1016/j.jspi.2010.10.003
  12. Tibshirani R (1996). Regression shrinkage and selection via the Lasso, Journal of the Royal Statistical Society Series B, 58, 267-288.
  13. Wahba G (1983). Bayesian confidence interval for the cross-validated smoothing spline, Journal of the Royal Statistical Society Series B, 45, 133-150.
  14. Xia Y and Li WK (1999). On single-index coefficient regression models, Journal of the American Statistical Association, 94, 1275-1285. https://doi.org/10.1080/01621459.1999.10473880
  15. Xue LG and Wang QH (2012). Empirical likelihood for single-index varying-coefficient models, Bernoulli, 18, 836-856. https://doi.org/10.3150/11-BEJ365
  16. Yang H, Guo C, and Lv J (2014). A robust and efficient estimation method for single-index varyingcoefficient models, Statistics and Probability Letters, 94, 119-127. https://doi.org/10.1016/j.spl.2014.07.011
  17. Yang H and Yang J (2014). The adaptive L1-penalized LAD regression for partially linear singleindex models, Journal of Statistical Planning and Inference, 151, 73-89. https://doi.org/10.1016/j.jspi.2014.04.003
  18. Zhu H, Lv Z, Yu K, and Deng C (2015). Robust variable selection in partially varying coefficient single-index model, Journal of the Korean Statistical Society, 44, 45-57. https://doi.org/10.1016/j.jkss.2014.05.002
  19. Zou H (2006). The adaptive LASSO and its oracle properties, Journal of the American Statistical Association, 101, 1418-1429. https://doi.org/10.1198/016214506000000735