Asymptotic Results for an M-Estimator of the Regression Function for Quasi-Associated Processes

Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

In this paper, we study a family of robust nonparametric estimators for the regression function based on the kernel method. It is assumed that the observations form a stationary quasi-associated sequence. Under general conditions we establish the almost-complete convergence with rate of the estimator as well as its asymptotic normality.

References

  1. 1.
    Barlow, R.E., Proschan, F.: Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart & Winston, New York (1981)Google Scholar
  2. 2.
    Boente, G., Fraiman, R.: Robust nonparametric regression estimation. J. Multivar. Anal. 29, 180–198 (1989)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Boente, G., Fraiman, R.: Asymptotic distribution of robust estimators for nonparametric models from mixing processes. Ann. Stat. 18, 891–906 (1990)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Boente, G., Ganzalez-Manteiga, W., Pérez-Gonzalez, A.: Robust nonparametric estimation with missing data. J. Stat. Plann. Inference 139, 571–592 (2009)MATHCrossRefGoogle Scholar
  5. 5.
    Bulinski, A., Shabanovich, E.: Asymptotical behaviour of some functionals of positively and negatively dependent random fields (in Russian). Fundam. Appl. Math. 4, 479–492 (1998)Google Scholar
  6. 6.
    Bulinski, A., Suquet, C.: Normal approximation for quasi-associated random fields. Stat. Probab. Lett. 54, 215–226 (2001)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Collomb, G., Härdle, W.: Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations. Stoch. Process. Appl. 23, 77–89 (1986)MATHCrossRefGoogle Scholar
  8. 8.
    Dedecker, J., Doukhan, P., Lang, G., León, R.J.R., Louhichi, S., Prieur, C.: Weak Dependence: With Examples and Applications. Lecture Notes in Statistics. Springer, New York (2007)CrossRefGoogle Scholar
  9. 9.
    Douge, L.: Théorèmes limites pour des variables quasi-associées hilbertiennes. Ann. Inst. Stat. Univ. Paris 54, 51–60 (2010)MathSciNetGoogle Scholar
  10. 10.
    Doukhan, P., Lang, G., Surgailis, D., Tyssière, G.: Dependence in Probability and Statistics. Lecture Notes in Statistics. Springer, Berlin (2010)MATHCrossRefGoogle Scholar
  11. 11.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)MATHGoogle Scholar
  12. 12.
    Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Stat. 38, 1466–1476 (1967)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Györfi, L., Härdle, W., Sarda, P., Vieu, P.: Nonparametric Curve Estimation from Time Series. Lecture Notes in Statistics, vol. 60. Springer, New York (1989)Google Scholar
  14. 14.
    Härdle, W., Tsybakov, A.: Robust nonparametric regression with simultaneous scale curve estimation. Ann. Stat. 16, 120–135 (1988)MATHCrossRefGoogle Scholar
  15. 15.
    Huber, P.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)MATHCrossRefGoogle Scholar
  16. 16.
    Jong-Dev, K., Proschan, F.: Negative association of random variables, with applications. Ann. Stat. 11, 286–295 (1983)CrossRefGoogle Scholar
  17. 17.
    Kallabis, R.S., Neumann, M.H.: An exponential inequality under weak dependence. Bernoulli 12, 333–350 (2006)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Laïb, N., Ould Saïd, E.: A robust nonparametric estimation of the autoregression function under an ergodic hypothesis. Can. J. Stat. 28, 817–828 (2000)MATHCrossRefGoogle Scholar
  19. 19.
    Masry, E.: Recursive probability density estimation for weakly dependent stationary processes IEEE. Trans. Inform. Theory 32, 254–267 (1986)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Masry, E.: Multivariate probability density estimation for associated processes: strong consistency and rates. Stat. Probab. Lett. 58, 205–219 (2002)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Matula, P.: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 15, 209–213 (1992)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Robinson, P.M.: Robust nonparametric autoregression. In: Robust and Nonlinear Time Series Analysis (Heidelberg, 1983). Lectures Note in Statistics, vol. 26, pp. 247–255. Springer, New York (1983)Google Scholar
  23. 23.
    Roussas, G.G.: Kernel estimates under association: strong uniform consistency. Stat. Probab. Lett. 12, 215–224 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Roussas, G.G.: Curve estimation in random fields of associated processes. J. Nonparametric Stat. 2, 215–224 (1993)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Roussas, G.G.: Prediction under association. In: Limnios, N., Nikulin, M.S. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference. Birkhäuser, Boston (2000)Google Scholar
  26. 26.
    Shashkin, A.P.: Quasi-association for system of Gaussian vectors. Usp. Mat. Nauk (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Djillali LiabèsSidi Bel AbbèsAlgerie
  2. 2.Département de MathématiquesUniversité des Sciences et de la Technologie, Mohamed BoudiafEl Mnaouer-OranAlgerie
  3. 3.Laboratoire de Statistique et Processus StochastiquesUniversité Djillali LiabèsSidi Bel AbbèsAlgerie
  4. 4.Université Lille Nord de FranceLilleFrance
  5. 5.ULCO, LMPACalaisFrance

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