Skip to main content
SpringerLink
Log in

Rank-Based Test for Partial Functional Linear Regression Models

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper investigates the hypothesis test of the parametric component in partial functional linear regression models. Based on a rank score function, the authors develop a rank test using functional principal component analysis, and establish the asymptotic properties of the resulting test under null and local alternative hypotheses. A simulation study shows that the proposed test procedure has good size and power with finite sample sizes. The authors also present an illustration through fitting the Berkeley Growth Data and testing the effect of gender on the height of kids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cuevas A, A partial overview of the theory of statistics with functional data, Journal of Statistical Planning and Inference, 2014, 147: 1–23.

    Article  MathSciNet  Google Scholar 

  2. Horváth L and Kokoszka P, Inference for Functional Data with Applications, Springer, New York, 2012.

    Book  Google Scholar 

  3. Cardot H, Ferraty F, and Sarda P, Spline estimators for the functional linear model, Statistica Sinica, 2003, 13(3): 571–592.

    MathSciNet  MATH  Google Scholar 

  4. Yao F, Müller H G, and Wang J L, Functional linear regression analysis for longitudinal data, The Annals of Statistics, 2005, 33(6): 2873–2903.

    Article  MathSciNet  Google Scholar 

  5. Crambes C, Kneip A, and Sarda P, Smoothing splines estimators for functional linear regression, The Annals of Statistics, 2009, 37(1): 35–72.

    Article  MathSciNet  Google Scholar 

  6. Hall P and Horowitz J L, Methodology and convergence rates for functional linear regression, The Annals of Statistics, 2007, 35(1): 70–91.

    Article  MathSciNet  Google Scholar 

  7. Cai T and Hall P, Prediction in functional linear regression, The Annals of Statistics, 2006, 34(5): 2159–2179.

    Article  MathSciNet  Google Scholar 

  8. Yuan M and Cai T, A reproducing kernel Hilbert space approach to functional linear regression, The Annals of Statistics, 2010, 38(6): 3412–3444.

    Article  MathSciNet  Google Scholar 

  9. Shin H, Partial functional linear regression, Journal of Statistical Planning and Inference, 2009, 139(10): 3405–3418.

    Article  MathSciNet  Google Scholar 

  10. Kong D, Xue K, Yao F, et al., Partially functional linear regression in high dimensions, Biometrika, 2016, 103(1): 147–159.

    Article  MathSciNet  Google Scholar 

  11. Yu P, Zhang Z Z, and Du J, A test of linearity in partial functional linear regression, Metrika, 2016, 79(8): 953–969.

    Article  MathSciNet  Google Scholar 

  12. Aneiros-Pérez G and Vieu P, Automatic estimation procedure in partial linear model with functional data, Statistical Papers, 2011, 52(4): 751–771.

    Article  MathSciNet  Google Scholar 

  13. Peng Q Y, Zhou J J, and Tang N S, Varying coefficient partially functional linear regression models, Statistical Papers, 2016, 57: 827–841.

    Article  MathSciNet  Google Scholar 

  14. Yu P, Du J, and Zhang Z Z, Single-index partially functional linear regression model, Statistical Papers, 2018, (11): 1–17.

    Google Scholar 

  15. Hettmansperger T P and McKean J W, Robust Nonparametric Statistical Methods, 2nd Edition, CRC Press, Boca Raton, FL, USA, 2011.

    MATH  Google Scholar 

  16. Leng C, Variable selection and coefficient estimation via regularized rank regression, Statistica Sinica, 2010, 20: 167–181.

    MathSciNet  MATH  Google Scholar 

  17. Du J, Chen X P, Kwessi E, et al., Model averaging based on rank, Journal of Applied Statistics, 2018, 45(10): 1900–1919.

    Article  MathSciNet  Google Scholar 

  18. Wang L, Kai B, and Li R, Local rank inference for varying coefficient models, Journal of the American Statistical Association, 2009, 104(488): 1631–1645.

    Article  MathSciNet  Google Scholar 

  19. Liu A Y, Li Q Z, Liu C L, et al., A rank-based test for comparison of multidimensional outcomes, Journal of the American Statistical Association, 2010, 105(490): 578–587.

    Article  MathSciNet  Google Scholar 

  20. Li Z B, Cao F, Zhang J J, et al., Summation of absolute value test for multiple outcome comparison with moderate effect, Journal of Systems Science & Complexity, 2013, 26(3): 462–469.

    Article  MathSciNet  Google Scholar 

  21. Sun J and Lin L, Local rank estimation and related test for varying-coefficient partially linear models, Journal of Nonparametric Statistics, 2014, 26(1): 187–206.

    Article  MathSciNet  Google Scholar 

  22. Feng L, Wang Z J, Zhang C, et al., Nonparametric testing in regression models with Wilcoxontype generalized likelihood ratio, Statistica Sinica, 2016, 26: 137–155.

    MathSciNet  Google Scholar 

  23. Yang J, Yang H, and Lu F, Rank-based shrinkage estimation for identification in semiparametric additive models, Statistical Papers, 2019, 60(4): 1255–1281.

    Article  MathSciNet  Google Scholar 

  24. Zhao W, Lian H, and Ma S, Robust reduced-rank modeling via rank regression, Journal of Statistical Planning and Inference, 2014, 180: 1–12.

    Article  MathSciNet  Google Scholar 

  25. Bindele H F and Zhao Y C, Rank-based estimating equation with non-ignorable missing responses via empirical likelihood, Statistica Sinica, 2018, 28(4): 1787–1820.

    MathSciNet  MATH  Google Scholar 

  26. Bindele H F, Abebe A, and Meyer K N, General rank-based estimation for regression single index models, Annals of the Institute of Statistical Mathematics, 2018, 70: 1115–1146.

    Article  MathSciNet  Google Scholar 

  27. Sun W, Bindele H F, Abebe A, et al., General local rank estimation for single-index varying coefficient models, Journal of Statistical Planning and Inference, 2019, https://doi.org/10.1016/j.jspi.2019.01.005.

    Google Scholar 

  28. Hettmansperger T P, Statistical Inference Based on Ranks, Wiley, New York, 1984.

    MATH  Google Scholar 

  29. Lu Y, Du J, and Sun Z M, Functional partially linear quantile regression model, Metrika, 2014, 77(3): 317–332.

    Article  MathSciNet  Google Scholar 

  30. Tuddenham R and Snyder M, Physical growth of California boys and girls from birth to eighteen years, California Publications on Child Development, 1954, 1: 183–364.

    Google Scholar 

  31. Ramsay J O, Bock R D, and Gasser T, Comparison of height acceleration curves in the Fels, Zurich, and Berkeley growth data, Annals of Human Biology, 1995, 22(5): 413–426.

    Article  Google Scholar 

  32. Chiou J M and Li P L, Functional clustering and identifying substructures of longitudinal data, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2007, 69(4): 679–699.

    Article  MathSciNet  Google Scholar 

  33. Van der Vaart A W, Asymptotic Statistics, Cambridge University Press, Oxford, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Applied Sciences, Beijing, 100124, China

    Tianfa Xie & Ruiyuan Cao

  2. School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041000, China

    Ping Yu

Authors
  1. Tianfa Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruiyuan Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianfa Xie.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11771032, 11571340 and 11701020, the Science and Technology Project of Beijing Municipal Education Commission under Grant Nos. KM201710005032 and KM201910005015, and the International Research Cooperation Seed Fund of Beijing University of Technology under Grant No. 006000514118553.

This paper was recommended for publication by Editor LI Qizhai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xie, T., Cao, R. & Yu, P. Rank-Based Test for Partial Functional Linear Regression Models. J Syst Sci Complex 33, 1571–1584 (2020). https://doi.org/10.1007/s11424-020-8362-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-020-8362-2

Keywords

Search

Navigation