Skip to main content

Missing responses at random in functional single index model for time series data

Abstract

In this paper, we first investigate the estimation of the functional single index regression model with missing responses at random for strong mixing time series data. More precisely, the uniform almost complete convergence rate and asymptotic normality of the estimator are obtained respectively under some general conditions. Then, some simulation studies are carried out to show the finite sample performances of the estimator. Finally, a real data analysis about the sea surface temperature is used to illustrate the effectiveness of our methodology.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Ait-Saïdi A, Ferraty F, Kassa R, Vieu P (2008) Cross-validated estimation in the single functional index model. Statistics 42(6):475–494

    MathSciNet  Article  Google Scholar 

  2. Aneiros G, Bongiorno EG, Cao R, Vieu P (2017) Functional statistics and related fields. Springer, Contributions to Statistics

  3. Aneiros G, Cao R, Fraiman R, Genest C, Vieu P (2019) Recent advances in functional data analysis and high-dimensional statistics. J Multivar Anal 170:3–9

    MathSciNet  Article  Google Scholar 

  4. Attaoui S, Ling NX (2016) Asymptotic results of a nonparametric conditional cumulative distribution estimator in the single functional index modeling for time series data with applications. Metrika 79:485–511

    MathSciNet  Article  Google Scholar 

  5. Attaoui S, Laksaci A, Ould-Said E (2011) A note on the conditional density estimate in the single functional index model. Stat Probab Lett 81(1):45–53

    MathSciNet  Article  Google Scholar 

  6. Attouch M, Laksaci A, Ould-Said E (2010) Asymptotic normality of a robust estimator of the regression function for functional time series data. J Korean Stat Soc 39:489–500

    MathSciNet  Article  Google Scholar 

  7. Bosq D (2000) Linear processes in function spaces: theory and applications. Lecture notes in statistics. Springer, New York

    Book  Google Scholar 

  8. Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plan Inference 147:1–23

    MathSciNet  Article  Google Scholar 

  9. Cheng PE (1994) Nonparametric estimation of mean functionals with data missing at random. J Am Stat Assoc 89:81–87

    Article  Google Scholar 

  10. Davydov Y (1970) The invariance principle for stationary processes. Theory Probab Appl 15:487–498

    MathSciNet  Article  Google Scholar 

  11. Delsol L (2009) Advances on asymptotic normality in non-parametric functional time series analysis. Statistics 43:13–33

  12. Doukhan P (1994) Mixing: properties and examples. Lecture notes in statistics, vol 85. Springer, Berlin

    Book  Google Scholar 

  13. Ding H, Liu YH, Xu WC, Zhang RQ (2017) A class of functional partially linear single-index models. J Multivar Anal 161:68–82

    MathSciNet  Article  Google Scholar 

  14. Efromovich S (2011) Nonparametric regression with predictors missing at random. J Am Stat Assoc 106:306–319

    MathSciNet  Article  Google Scholar 

  15. Febrero-Bande M, Galeano P, Gonzalez-Manteiga W (2019) Estimation imputation and prediction for the functional linear model with scalar response with responses missing at random. Comput Stat Data Anal 131:91–103

    MathSciNet  Article  Google Scholar 

  16. Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Theory and practice. Springer

  17. Ferraty F, Peuch A, Vieu P (2003) Modèle à indice fonctionnel simple. C R Math Paris 336(12):1025–1028

    MathSciNet  Article  Google Scholar 

  18. Ferraty F, Rabhi A, Vieu P (2005) Conditional quantiles for dependent functional data with application to the climatic El Nino Phenomenon. Indian J Stat 67:378–398

    MathSciNet  MATH  Google Scholar 

  19. Ferraty F, Laksaci A, Tadj A, Vieu P (2010) Rate of uniform consistency for nonparametric estimates with functional variables. J Stat Plan Inference 140:335–352

    MathSciNet  Article  Google Scholar 

  20. Ferraty F, Sued M, Vieu P (2013) Mean estimation with data missing at random for functional covariables. Statistics 47(4):688–706

    MathSciNet  Article  Google Scholar 

  21. Goia A, Vieu P (2016) An introduction to recent advances in high. Infinite dimensional statistics. J Multivar Anal 146:1–6

    MathSciNet  Article  Google Scholar 

  22. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New York

    Book  Google Scholar 

  23. Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis with an introduction to linear operators. Wiley series in probability and statistics

  24. Kraus D (2019) Inferential procedures for partially observed functional data. J Multivar Anal 173:583–603

    MathSciNet  Article  Google Scholar 

  25. Liang H, Wang S, Carroll R (2007) Partially linear models with missing response variables and error-prone covariates. Biometrika 94:185–198

    MathSciNet  Article  Google Scholar 

  26. Liebscher E (2001) Central limit theorems for-mixing triangular arrays with application to nonparametric statistics. Mathematical Methods of Statistics 10:194–214

  27. Ling NX, Li ZH (2014) Conditional density estimation in the single functional index model for \( \alpha \)-mixing functional data. Commun Stat Theory Methods 43:441–454

    MathSciNet  Article  Google Scholar 

  28. Ling NX, Vieu P (2018) Nonparametric modelling for functional data: selected survey and tracks for future. Statistics 52(4):934–949

    MathSciNet  Article  Google Scholar 

  29. Ling NX, Xu Q (2012) Asymptotic normality of conditional density estimation in the single index model for functional time series data. Stat Probab Lett 82:2235–2243

    MathSciNet  Article  Google Scholar 

  30. Ling NX, Kan R, Vieu P, Meng SY (2019) Semi-functional partially linear regression model with responses missing at random. Metrika 82:39–70

  31. Ling NX, Liang LL, Vieu P (2015) Nonparametric regression estimation for functional stationary ergodic data with missing at random. J Stat Plan Inference 162:75–87

    MathSciNet  Article  Google Scholar 

  32. Little R, Rubin D (2002) Statistical analysis with missing data, 2nd edn. Wiley, New York

    Book  Google Scholar 

  33. Nittner T (2003) Missing at random (MAR) in nonparametric regression: a simulation experiment. Stat Methods Appl 12(2):195–210

    MathSciNet  Article  Google Scholar 

  34. Ramsay J, Silverman B (2005) Functional data analysis. Springer, Springer

    Book  Google Scholar 

  35. Wang QH, Linton O, Wolfgang H (2004) Semiparametric regression analysis with missing response at random. J Am Stat Assoc 99:334–345

    MathSciNet  Article  Google Scholar 

  36. Wu JW, Peng HX, Tu WZ (2019) Large-sample estimation and inference in multivariate single-index models. J Multivar Anal 171:382–396

    MathSciNet  Article  Google Scholar 

  37. Xue LG, Zhu LX (2010) Empirical likelihood in nonparametric and semiparametric model. Science Press, Beijing

    Google Scholar 

  38. Yu P, Du J, Zhang ZZ (2018) Single-index partially functional linear regression model. Statistical papers

Download references

Acknowledgements

The authors would like to thank the Editor in Chief, the A.E and the two anonymous reviewers for their insightful comments and suggestions, which have led to a great improvement of this present version. This research is supported by the National Natural Science Foundation of China (Grant Nos.: 72071068, 11901286), and Vieu’s research is partially supported by Spanish Ministerio de Economíay Competitividad (Grant No.: MTM2014-52876-R), which are acknowledged.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nengxiang Ling.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ling, N., Cheng, L., Vieu, P. et al. Missing responses at random in functional single index model for time series data. Stat Papers (2021). https://doi.org/10.1007/s00362-021-01251-2

Download citation

Keywords

  • Functional single index model
  • Uniform almost complete convergence rate
  • Asymptotic normality
  • Strong mixing dependence
  • Missing responses at random