Advertisement

Statistics and Computing

, Volume 27, Issue 2, pp 571–582 | Cite as

Penalised spline estimation for generalised partially linear single-index models

  • Yan Yu
  • Chaojiang WuEmail author
  • Yuankun Zhang
Article

Abstract

Generalised linear models are frequently used in modeling the relationship of the response variable from the general exponential family with a set of predictor variables, where a linear combination of predictors is linked to the mean of the response variable. We propose a penalised spline (P-spline) estimation for generalised partially linear single-index models, which extend the generalised linear models to include nonlinear effect for some predictors. The proposed models can allow flexible dependence on some predictors while overcome the “curse of dimensionality”. We investigate the P-spline profile likelihood estimation using the readily available R package mgcv, leading to straightforward computation. Simulation studies are considered under various link functions. In addition, we examine different choices of smoothing parameters. Simulation results and real data applications show effectiveness of the proposed approach. Finally, some large sample properties are established.

Keywords

Generalised linear model Generalised additive model Low rank approximation Penalised splines  Profile likelihood 

Notes

Acknowledgments

We thank the Editor for suggesting the profile likelihood approach and for many extremely valuable points. This paper would not be possible otherwise. We also thank an anonymous referee for the thoughtful comments. Finally, we thank Xia, Y. and Härdle W. for providing the French Bank dataset.

Supplementary material

11222_2016_9639_MOESM1_ESM.pdf (1022 kb)
Supplementary material 1 (pdf 1022 KB)

References

  1. Anderssen, R.S., Bloomfield, P.: A time series approach to numerical differentiation. Technometrics 16, 69–75 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Boente, G., Rodriguez, D.: Robust estimates in generalised partially linear single-index models. Test 21(2), 386–411 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Carroll, R.J., Fan, J., Gijbels, I., Wand, M.P.: Generalized partially linear single-index models. J. Am. Stat. Assoc. 92, 477–489 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Carroll, R.J., Ruppert, D., Stefanski, L.A., Crainiceanu, C.M.: Measurement Error in Nonlinear Models: A Modern Perspective. CRC press, Boca Raton (2012)zbMATHGoogle Scholar
  5. Crainiceanu, C.M., Ruppert, D., Wand, M.P.: Bayesian analysis for penalized spline regression using WinBUGS. J. Stat. Softw. 14(14), 1–24 (2005)CrossRefGoogle Scholar
  6. Craven, P., Wahba, G.: Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377–403 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11, 89–121 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gray, R.J.: Spline-based tests in survival analysis. Biometrics 50, 640–652 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Härdle, W., Hall, P., Ichimura, H.: Optimal smoothing in single-index models. Ann. Stat. 21, 157–178 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hastie, T.J., Tibshirani, R.: Generalized Additive Models. Chapman & Hall, London (1990)zbMATHGoogle Scholar
  11. Huh, J., Park, B.U.: Likelihood-based local polynomial fitting for single-index models. J. Multivar. Anal. 80, 302–321 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Liang, H., Liu, X., Li, R., Tsai, C.: Estimation and testing for partially linear single-index models. Ann. Stat. 38, 3811–3836 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman & Hall, London (1989)CrossRefzbMATHGoogle Scholar
  14. Parker, R.L., Rice, J.A.: Discussion of Some aspects of the spline smoothing approach to non-parametric regression curve fitting by B. W. Silverman. J. R. Stat. Soc. Ser. B (Methodol.) 47, 1–52 (1985)Google Scholar
  15. Poon, W.Y., Wang, H.B.: Bayesian analysis of generalized partially linear single-index models. Comput. Stat. Data Anal. 68, 251–261 (2013)MathSciNetCrossRefGoogle Scholar
  16. Qu, A., Li, R.: Quadratic inference functions for varying-coefficient models with longitudinal data. Biometrics 62, 379–391 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Ruppert, D., and Carroll, R.: Penalized Regression Splines, working paper, Cornell University, School of Operations Research and Industrial Engineering (1997). www.orie.cornell.edu/davidr/papers
  18. Ruppert, D.: Selecting the number of knots for penalized splines. J. Comput. Graph. Stat. 11, 735–757 (2002)MathSciNetCrossRefGoogle Scholar
  19. Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric Regression. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  20. Ruppert, D., Carroll, R.: Spatially-adaptive penalties for spline fitting. Aust. N. Z. J. Stat. 42, 205–223 (2000)CrossRefGoogle Scholar
  21. Wahba, G.: A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Stat. 13(4), 1378–1402 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Wood, S.N.: Modelling and smoothing parameter estimation with multiple quadratic penalties. J. R. Stat. Soc. Ser. B 62(2), 413–428 (2000)MathSciNetCrossRefGoogle Scholar
  23. Wood, S.N.: Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Am. Stat. Assoc. 99, 673–686 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wood, S.N.: Generalized Additive Models: An Introduction with R. CRC Chapman and Hall, Boca Raton (2006)zbMATHGoogle Scholar
  25. Wood, S.N.: Fast stable direct fitting and smoothness selection for generalized additive models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 70(3), 495–518 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wood, S.N.: Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73(1), 3–36 (2011)MathSciNetCrossRefGoogle Scholar
  27. Xia, Y., Härdle, W.: Semi-parametric estimation of partially linear single index models. J. Multivar. Anal. 97, 1162–1184 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yi, G.Y., He, W., Liang, H.: Analysis of correlated binary data under partially linear single-index logistic models. J. Multivar. Anal. 100(2), 278–290 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Yu, Y., Ruppert, D.: Penalized spline estimation for partially linear single-index models. J. Am. Stat. Assoc. 97, 1042–1054 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Operations, Business Analytics and Information SystemsUniversity of CincinnatiCincinnatiUSA
  2. 2.Department of Decision Sciences and MISDrexel UniversityPhiladelphiaUSA
  3. 3.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations