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Smooth Estimation of Error Distribution in Nonparametric Regression Under Long Memory

  • Hira L. KoulEmail author
  • Lihong Wang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 193)

Abstract

We consider the problem of estimating the error distribution function in a nonparametric regression model with long memory design and long memory errors. This paper establishes a uniform reduction principle of a smooth weighted residual empirical distribution function estimator. We also investigate consistency property of local Whittle estimator of the long memory parameter based on nonparametric residuals. The results obtained are useful in providing goodness of fit test for the marginal error distribution and in prediction under long memory.

Keywords

Kernel estimation Uniform reduction principle 

Notes

Acknowledgements

Research of Hira L. Koul was in part supported by the NSF-DMS grant 1205271. Research of Lihong Wang was in part supported by NSFC Grants 11671194 and 11171147. Authors are grateful to a delegant referee whose comments helped to improve the presentation and some of the proofs.

References

  1. 1.
    Csörgo, S., Mielniczuk, J.: Nonparametric regression under long-range dependent normal errors. Ann. Statist. 23, 1000–1014 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dalla, V., Giraitis, L., Hidalgo, J.: Consistent estimation of the memory parameter for nonlinear time series. J. Time Ser. Anal. 27, 211–251 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fernholz, L.: Almost sure convergence of smoothed empirical distribution functions. Scand. J. Statist. 18, 255–262 (1991)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Giraitis, L., Koul, H.L., Surgailis, D.: Asymptotic normality of regression estimators with long memory errors. Statist. Probab. Lett. 29, 317–335 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Giraitis, L., Koul, H.L., Surgailis, D.: Large sample inference for long memory processes. Imperial College Press, London (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Guo, H., Koul, H.: Asymptotic inference in some heteroscedastic regression models with long memory design and errors. Ann. Statist. 36, 458–487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Koul, H.L., Mimoto, N., Surgailis, D.: Goodness-of-fit tests for long memory moving average marginal density. Metrika 76, 205–224 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Koul, H.L., Surgailis, D.: Asymptotic expansion of the empirical process of long memory moving averages. In: Dehling, H., Mikosch, T., Sørensen, M. (eds.) Empirical Process Techniques for Dependent Data, pp. 213–239. Birkhäser, Boston (2002)Google Scholar
  9. 9.
    Koul, H.L., Surgailis, D.: Goodness-of-fit testing under long memory. J. Statist. Plann. Inference 140, 3742–3753 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Koul, H.L., Surgailis, D.: Goodness-of-fit tests for marginal distribution of linear random fields with long memory. RM # 702, Department of Statistics & Probability, Michigan State University (2013)Google Scholar
  11. 11.
    Kulik, R., Wichelhaus, C.: Nonparametric conditional variance and error density estimation in regression models with dependent errors and predictors. Electron. J. Stat. 5, 856–898 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, R., Yang, L.: Kernel estimation of multivariate cumulative distribution function. J. Nonparametr. Stat. 20, 661–677 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lorek, P., Kulik, R.: Empirical process of residuals for regression models with long memory errors. Statist. Probab. Lett. 86, 7–16 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Masry, E., Mielniczuk, J.: Local linear regression estimation for time series with long-range dependence. Stochastic Process. Appl. 82, 173–193 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Müller, U.U., Schick, A., Wefelmeyer, W.: Estimating the innovation distribution in nonparametric autoregression. Probab. Theory Relat. Fields 144, 53–77 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Robinson, P.M.: Semiparametric analysis of long memory time series. Ann. Statist. 22, 515–539 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Robinson, P.M.: Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23, 1630–1661 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Robinson, P.M.: Large-sample inference for nonparametric regression with dependent errors. Ann. Statist. 25, 2054–2083 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robinson, P.M.: Asymptotic theory for nonparametric regression with spatial data. J. Econometrics 165, 5–19 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, J., Liu, R.: Cheng, F., Yang, L.: Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band. Ann. Statist. 42, 654–668 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zygmund, A.: Trigonometric series, 3rd edn. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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