Robust exponential squared loss-based estimation in semi-functional linear regression models

  • Ping Yu
  • Zhongyi Zhu
  • Zhongzhan Zhang
Original Paper


In this paper, we present a new robust estimation procedure for semi-functional linear regression models by using exponential squared loss. The outstanding advantage of the proposed method is the resulting estimators are more efficient than the least squares estimators in the presence of outliers or heavy-tail error distributions. The slope function and functional predictor variable are approximated by functional principal component basis functions. Under some regularity conditions, we obtain the optimal convergence rate of slope function, and the asymptotic normality of parameter vector and variance estimator. Finally, we investigate the finite sample performance of the proposed method through a simulation study and real data analysis.


Functional data analysis Functional principal component analysis Exponential squared loss Robust estimation 


  1. Aneiros-Pérez G, Ling N, Vieu P (2015) Error variance estimation in semi-functional partially linear regression models. J Nonparametr Stat 27(3):316–330MathSciNetCrossRefzbMATHGoogle Scholar
  2. Aneiros-Pérez G, Raña P, Vieu P, Vilar P (2017) Bootstrap in semi-functional partial linear regression under dependence. TEST 2017:1–21Google Scholar
  3. Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76(11):1102–1110MathSciNetCrossRefzbMATHGoogle Scholar
  4. Aneiros-Pérez G, Vieu P (2013) Testing linearity in semi-parametric functional data analysis. Comput Stat 28:413–434MathSciNetCrossRefzbMATHGoogle Scholar
  5. Aneiros-Pérez G, Vieu P (2015) Partial linear modelling with multi-functional covariates. Comput Stat 30(3):647–671MathSciNetCrossRefzbMATHGoogle Scholar
  6. Brunel É, Mas A, Roche A (2016) Non-asymptotic adaptive prediction in functional linear models. J Multivar Anal 143:208–232MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cai T, Hall P (2006) Prediction in functional linear regression. Ann Stat 34(5):2159–2179MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cai T, Yuan M (2012) Minimax and adaptive prediction for functional linear regression. J Am Stat Assoc 107(499):1201–1216MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cardot H, Ferraty F, Sarda P (1999) Functional linear model. Stat Probab Lett 45(1):11–22MathSciNetCrossRefzbMATHGoogle Scholar
  10. Crambes C, Kneip A, Sarda P (2009) Smoothing splines estimators for functional linear regression. Ann Stat 37(1):35–72MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ferraty F, Goia A, Salinelli E, Vieu P (2013) Functional projection pursuit regression. TEST 22(2):293–320MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, New YorkzbMATHGoogle Scholar
  13. Goia A, Vieu P (2014) Some advances on semi-parametric functional data modelling. In: Contributions in infinite-dimensional statistics and related topics, Esculapio, BolognaGoogle Scholar
  14. Goia A, Vieu P (2015) A partitioned single functional index model. Comput Stat 30(3):673–692MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hall P, Hooker G (2016) Truncated linear models for functional data. J R Stati Soc Ser B (Stat Methodol) 78(3):637–653MathSciNetCrossRefGoogle Scholar
  16. Hall P, Horowitz JL (2007) Methodology and convergence rates for functional linear regression. Ann Stat 35(1):70–91MathSciNetCrossRefzbMATHGoogle Scholar
  17. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  18. Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  19. Huber P (1981) Robust estimation. Wiley, New YorkzbMATHGoogle Scholar
  20. Imaizumi M, Kato K (2018) PCA-based estimation for functional linear regression with functional responses. J Multivar Anal 163:15–36MathSciNetCrossRefzbMATHGoogle Scholar
  21. Jiang Y, Ji Q, Xie B (2017) Robust estimation for the varying coefficient partially nonlinear models. J Comput Appl Math 326:31–43MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kai B, Li R, Zou H (2011) New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann Stat 39(1):305–332MathSciNetCrossRefzbMATHGoogle Scholar
  23. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kokoszka P, Reimherr M (2017) Introduction to functional data analysis. CRC Press, Boca RatonzbMATHGoogle Scholar
  25. Kong D, Xue K, Yao F, Zhang H (2016) Partially functional linear regression in high dimensions. Biometrika 103(1):147–159MathSciNetCrossRefzbMATHGoogle Scholar
  26. Lin Z, Cao J, Wang L, Wang H (2017) Locally sparse estimator for functional linear regression models. J Comput Graph Stat 26(2):306–318MathSciNetCrossRefGoogle Scholar
  27. Ling N, Aneiros G, Vieu P (2017) kNN estimation in functional partial linear modeling. Stat Pap 1–22Google Scholar
  28. Lovric M (2011) International encyclopedia of statistical science. Springer, New YorkCrossRefzbMATHGoogle Scholar
  29. Lu Y, Du J, Sun Z (2014) Functional partially linear quantile regression model. Metrika 77(2):317–332MathSciNetCrossRefzbMATHGoogle Scholar
  30. Lv J, Yang H, Guo C (2015) Robust smooth-threshold estimating equations for generalized varying-coefficient partially linear models based on exponential score function. J Comput Appl Math 280:125–140MathSciNetCrossRefzbMATHGoogle Scholar
  31. Müller HG, Stadtmüller U (2005) Generalized functional linear models. Ann Stat 32(2):774–805MathSciNetCrossRefzbMATHGoogle Scholar
  32. Peng QY, Zhou JJ, Tang NS (2016) Varying coefficient partially functional linear regression models. Stat Pap 57(3):827–841MathSciNetCrossRefzbMATHGoogle Scholar
  33. Ramsay JO, Dalzell CJ (1991) Some tools for functional data analysis. J R Stat Soc Ser B (Methodol) 53(3):539–572MathSciNetzbMATHGoogle Scholar
  34. Ramsay JO, Silverman BW (2002) Applied functional data analysis: methods and case studies. Springer, New YorkCrossRefzbMATHGoogle Scholar
  35. Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  36. Shin H (2009) Partial functional linear regression. J Stat Plan Inference 139(10):3405–3418MathSciNetCrossRefzbMATHGoogle Scholar
  37. Song Y, Jian L, Lin L (2016) Robust exponential squared loss-based variable selection for high-dimensional single-index varying-coefficient model. J Comput Appl Math 308:330–345MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wang K, Lin L (2016) Robust structure identification and variable selection in partial linear varying coefficient models. J Stat Plan Inference 174:153–168MathSciNetCrossRefzbMATHGoogle Scholar
  39. Wang X, Jiang Y, Huang M, Zhang H (2013) Robust variable selection with exponential squared loss. J Am Stat Assoc 108(502):632–643MathSciNetCrossRefzbMATHGoogle Scholar
  40. Yao F, Müller HG, Wang JL (2005) Functional linear regression analysis for longitudinal data. Ann Stat 33(6):2873–2903MathSciNetCrossRefzbMATHGoogle Scholar
  41. Yu P, Zhang Z, Du J (2016) A test of linearity in partial functional linear regression. Metrika 79(8):953–969MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhou J, Chen Z, Peng Q (2016) Polynomial spline estimation for partial functional linear regression models. Comput Stat 31(3):1107–1129MathSciNetCrossRefzbMATHGoogle Scholar
  43. Zou H, Yuan M (2008) Composite quantile regression and the oracle model selection theory. Ann Stat 36(3):1108–1126MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsFudan UniversityShanghaiChina
  2. 2.School of Mathematics and Computer ScienceShanxi Normal UniversityLinfenChina
  3. 3.College of Applied SciencesBeijing University of TechnologyBeijingChina

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