Skip to main content
SpringerLink
Log in

Nonparametric M-estimation for Functional Stationary Ergodic Data

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper considers a nonparametric M-estimator of a regression function for functional stationary ergodic data. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. Under some mild conditions, the weak consistency and the asymptotic normality of the M-estimator are established. Furthermore, a simulated example is provided to examine the finite sample performance of the M-estimator.

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. Andrews, D.W.K. Non-strong mixing autoregressive processes. Journal of Applied Probability, 21(4): 930–934 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Azzedine, N., Laksaci, A., Ould- Saïd, E. On the robust nonparametric regression estimation for functional regressor. Statistics and Probability Letters, 78(18): 3216–3221 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Attouch, M., Laksaci, A., Ould- Saïd, E. Asymptotic distribution of robust estimator for functional nonparametric models. Communications in Statistics-Theory and Methods, 38(8): 1317–1335 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Attouch, M., Laksaci, A., Ould- Saïd, E. Asymptotic normality of a robust estimator of the regression function for functional time series data. Journal of the Korean Statistical Society, 39(4): 489–500 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.D., Rao, C.R., Wu, Y. M-estimation of multivariate linear regression parameters under a convex discrepancy function. Statistica Sinica, 2(1): 237–254 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Boente, G., Fraiman, R. Robust nonparametric regression estimation for dependent observations. Annals of Statistics, 17(3): 1242–1256 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bosq, D. Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed. Springer, Berlin, 1998

    Book  MATH  Google Scholar 

  8. Bosq, D. Linear processes in function spaces: Theory and applications. Springer, New York, 2000

    Book  MATH  Google Scholar 

  9. Cai, Z., Ould-Saïd, E. Local M-estimator for nonparametric time series. Statistics and Probability Letters, 65(4): 433–449 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, J., Zhang, L.X. Asymptotic properties of nonparametric M-estimation for mixing functional data. Journal of Statistical Planning and Inference, 139(2): 533–546 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chernick, M.R. A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Annals of Probability, 9(1): 145–149 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chobanyan, S.A., Tarieladze, V.l., Vakashnia, N.N. Probability Distributions on Banach Spaces. D. Reidel Publishing Company, Holland, 1987

    Google Scholar 

  13. Chow, Y.S., Teicher, H. Probability Theory. 2nd ed. Springer, Berlin, 1998

    MATH  Google Scholar 

  14. Cleveland, W.S. Robust locally weighted regression and smoothing scatter plots. Journal of the American Statistical Association, 74(368): 829–836 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cox, D.D. Asymptotics of M-type smoothing splines. Annals of Statistics, 11: 530–551 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crambes, C., Delsol, L., Laksaci, A. Robust nonparametric estimation for functional data. Journal of Nonparametric Statistics, 20(7): 573–598 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cunningham, J.K., Eubank, R.L., Hsing, T. M-type smoothing splines with auxiliary scale estimation. Computational Statistics and Data Analysis, 11(1): 43–51 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Davidson, J. Stochastic limit theory. Oxford University Press, New York, 1994

    Book  Google Scholar 

  19. Delsol, L. Advances on asymptotic normality in non-parametric functional time series analysis. Statistics, 43(1): 13–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fan, J., Hu, T., Truong, Y. Robust Non-parametric function estimation. Scandinavian Journal of Statistics, 21(4): 433–446 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Fan, J., Jiang, J. Variable bandwidth and one-step local M-estimator. Science China Mathematics, 43(1): 65–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferraty, F., Goia, A., Vieu, P. Functional nonparametric model for time series: a fractal approach for dimension reduction. Test, 11(2): 317–344 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferraty, F., Mas, A., Vieu, P. Nonparametric regression of functional data: inference and practical aspects. Australian and New Zealand Journal of Statistics, 49(3): 267–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ferraty, F., Vieu, P. Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normes. Comptes Rendus de l’Acaémie des Sciences-Series I-Mathematics, 330(2): 139–142 (2000)

    MATH  Google Scholar 

  25. Ferraty, F., Vieu, P. Nonparametric models for functional data, with applications in regression, time series prediction and curve discrimination. Journal of Nonparametric Statistics, 16(1–2): 111–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ferraty, F., Vieu, P. Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York, 2006

    MATH  Google Scholar 

  27. Gheriballah, A., Laksaci, A., Sekkal, S. Nonparametric M-regression for functional ergodic data. Statistics and Probability Letters, 83(3): 902–908 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. He, X.M., Zhu, Z.Y., Fung, W.K. Estimation in a semi-parametric model for longitudinal data with unspecified dependence structure. Biometrika, 89(3): 579–590 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hall, P., Heyde, C. Martingale Limit Theory and its Application. Academic Press, New York, 1980

    MATH  Google Scholar 

  30. Härdle, W. Robust regression function estimation. Journal of Multivariate Analysis, 14(2): 169–180 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Laïb, N. Kernel estimates of the mean and the volatility functions in a nonlinear autoregressive model with ARCH errors. Journal of Statistical Planning and Inference, 134(1): 116–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Laïb, N., Louani, D. Nonparametric kernel regression estimation for functional stationary ergodic data: asymptotic properties. Journal of Multivariate Analysis, 101(10): 2266–2281 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin, Z.Y., Li, D.G., Chen, J. Local linear M-estimators in null recurrent time series. Statistica Sinica, 19(4): 1683–1703 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Lin, Z.Y., Li, D.G., Gao, J.T. Local linear M-estimation in nonparametric spatial regression. Journal of Time Series Analysis, 30(3): 286–314 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lu, Z.D., Tang, Q.G., Cheng, L.S. Estimating spatial quantile regression with functional coeffients: a robust semiparametric framework. Bernoulli, 20(1): 164–189 (2013)

    Article  MATH  Google Scholar 

  36. Masry, E. Nonparametric regression estimation for dependent functional data: asymptotic normality. Stochastic Processes and their Applications, 115(1): 155–177 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pollard, D. Asymptotics for least absolute deviation regression estimators. Econometric Theory, 7(2): 186–199 (1991)

    Article  MathSciNet  Google Scholar 

  38. Ramsay, J., Silverman, B.W. Functional Data Analysis. Springer, New York, 1997

    Book  MATH  Google Scholar 

  39. Ramsay, J., Silverman, B.W. Applied Functional Data Analysis: Methods and Case Studies. Springer, New York, 2002

    Book  MATH  Google Scholar 

  40. Rockafellar, R.T. Convex Analysis. Princeton University Press, Princeton, 1970

    Book  MATH  Google Scholar 

  41. Tang, Q.G., Cheng, L.S. M-estimation and B-spline approximation for varying coefficient models with longitudinal data. Journal of Nonparametric Statistics, 20(7): 611–625 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tang, Q.G., Cheng, L.S. Asymptotic normality of M-estimators for varying coefficient models with longitudinal data. Communications in Statistics-Theory and Methods, 38(9): 1422–1440 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Tsybakov, A.B. Robust reconstruction of functions by the local-approximation method. Problems of Information Transmission, 22(2): 69–84 (1986)

    MathSciNet  Google Scholar 

  44. Xiong, X.Z., Lin Z.Y. Empirical likelihood inference for nonparametric regression functions with functional stationary ergodic data. Communications in Statistics-Theory and Methods, 42(19): 3421–3431 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Xian-zhu Xiong

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Zheng-yan Lin

Authors
  1. Xian-zhu Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng-yan Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zheng-yan Lin.

Additional information

This work is supported by National Natural Science Foundation of China (No.11301084) and Natural Science Foundation of Fujian Province, China (No.2014J01010).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiong, Xz., Lin, Zy. Nonparametric M-estimation for Functional Stationary Ergodic Data. Acta Math. Appl. Sin. Engl. Ser. 35, 491–512 (2019). https://doi.org/10.1007/s10255-019-0826-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-019-0826-6

Keywords

2000 MR Subject Classification

Search

Navigation