Skip to main content
SpringerLink
Log in

Nonparametric regression under dependent errors with infinite variance

  • Nonparametric Regression
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We consider local least absolute deviation (LLAD) estimation for trend functions of time series with heavy tails which are characterised via a symmetric stable law distribution. The setting includes both causal stable ARMA model and fractional stable ARIMA model as special cases. The asymptotic limit of the estimator is established under the assumption that the process has either short or long memory autocorrelation. For a short memory process, the estimator admits the same convergence rate as if the process has the finite variance. The optimal rate of convergencen −2/5 is obtainable by using appropriate bandwidths. This is distinctly different from local least squares estimation, of which the convergence is slowed down due to the existence of heavy tails. On the other hand, the rate of convergence of the LLAD estimator for a long memory process is always slower thann −2/5 and the limit is no longer normal.

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

  • Adler, R. J., Feldman, R. E. and Gallagher, C. (1997). Analysing stable time series,A User’s Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions and Processes (eds. R. J. Adler, R. E. Feldman and M. Taqqu), Birkhäuser, Boston.

    Google Scholar 

  • Altman, N. S. (1990). Kernel smoothing of data with correlated errors,Journal of the American Statistical Association,85, 749–759.

    Article  MathSciNet  Google Scholar 

  • Chiu, C. K. and Marron, J. S. (1991). Comparison of two bandwidth selectors with dependent errors,The Annals of Statistics,19, 1906–1918.

    MathSciNet  Google Scholar 

  • Csörgő, S. and Mielniczuk, J. (1995). Nonparametric regression under long-range dependent normal errors,The Annals of Statistics,23, 1000–1014.

    MathSciNet  Google Scholar 

  • Davis, R. A., Knight, K. and Liu, J. (1992). M-estimation for autoregressions with infinite variances,Stochastic Processes and their Applications,40, 145–180.

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, J. and Hall, P. (1994). On curve estimation by minimizing mean absolute deviation and its implications,The Annals of Statistics,22, 867–885.

    MathSciNet  MATH  Google Scholar 

  • Fan, J., Hu, T.-C. and Truong, Y. K. (1994). Robust nonparametric function estimation,Scandinavian Journal of Statistics,21, 433–446.

    MathSciNet  MATH  Google Scholar 

  • Geluk, J. and de Haan, L. (1987). Regular variation, extensions and Tauberian theorems,CWI Tract,40.

  • Hall, P. and Hart, J. D. (1990). Nonparametric regression with long-range dependence,Stochastic Processes and their Applications,36, 339–351.

    Article  MathSciNet  MATH  Google Scholar 

  • Hall, P., Peng, L. and Yao, Q. (2002). Prediction and nonparametric estimation for time series with heavy tails,Journal of Time Series Analysis,23(3), 313–331.

    Article  MathSciNet  MATH  Google Scholar 

  • Hart, J. D. (1991). Kernel regression estimation with time series errors,Journal of the Royal Statistical Society. Series B,53, 173–187.

    MathSciNet  MATH  Google Scholar 

  • Hart, J. D. (1994). Automated kernel smoothing of dependent data by using time series crossing-validation,Journal of the Royal Statistical Society. Series B,56, 529–542.

    MathSciNet  MATH  Google Scholar 

  • Herrmann, E., Gasser, T. and Kneip, A. (1992). Choice of bandwidth for kernel regression when residuals are correlated,Biometrika,79(4), 783–795.

    Article  MathSciNet  MATH  Google Scholar 

  • Hsing, T. (1999). On the asymptotic distributions of partial sums of functionals of infinite-variance moving averages,The Annals of Probability,27, 1579–1599.

    Article  MathSciNet  MATH  Google Scholar 

  • Kasahara, Y. and Maejima, M. (1988). Weighted sums of i.i.d. random variables attracted to integrals of stable processes,Probability Theory and Related Fields,78, 75–96.

    Article  MathSciNet  MATH  Google Scholar 

  • Klüppelberg, C., and Mikosch, T. (1996). The integrated periodogram for stable processes,The Annals of Statistics,24(5), 1855–1879.

    Article  MathSciNet  MATH  Google Scholar 

  • Kokoszka, P. and Taqqu, M. S. (1995). Fractional ARIMA with stable innovations,Stochastic Processes and their Applications,60, 19–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Kokoszka, P. and Taqqu, M. S. (1996). Parameter estimation for infinite variance fractional ARIMA,The Annals of Statistics,24, 1880–1913.

    Article  MathSciNet  MATH  Google Scholar 

  • Koul, H. and Surgailis, D. (2001). Asymptotics of empirical processes of long memory moving averages with infinite variance,Stochastic Processes and their Applications,91, 309–336.

    Article  MathSciNet  MATH  Google Scholar 

  • Mallows, C. L. (1980). Some theory of nonlinear smoothers,The Annals of Statistics,8, 695–715.

    MathSciNet  MATH  Google Scholar 

  • Mikosch, T., Gadrich, T., Klüppelberg, C. and Alder, R. J. (1995). Parameter estimation for ARMA models with infinite variance innovations,The Annals of Statistics,23, 305–326.

    MathSciNet  MATH  Google Scholar 

  • Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators,Econometric Theory,7, 186–198.

    Article  MathSciNet  Google Scholar 

  • Ray, B. K. and Tsay, R. S. (1997). Bandwidth selection for kernel regression with long-range dependence,Biometrika,84, 791–802.

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S. I. (1997). Heavy tail modeling and teletraffic data (with discussion),The Annals of Statistics,25, 1805–1869.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, P. M. (1994). Rates of convergence and optimal bandwidth choice for long-range dependence,Probability Theory and Related Fields,99, 443–473.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, P. M. (1997). Large-sample inference for nonparametric regression with dependent errors,The Annals of Statistics,25, 2054–2083.

    Article  MathSciNet  MATH  Google Scholar 

  • Roussas, G. G., Tran, L. T. and Ioannides, O. A. (1992). Fixed design regression for time series: Asymptotic normality,Journal of Multivariate Analysis,40, 262–291.

    Article  MathSciNet  MATH  Google Scholar 

  • Samorodnitsky, G. and Taqqu, M. S. (1994).Stable Non-Gaussian Random Processes, Chapman and Hall, New York.

    MATH  Google Scholar 

  • Surgailis, D. (2002). Stable limits of empirical processes of moving averages with infinite variance,Stochastic Processes and their Applications,100, 255–274.

    Article  MathSciNet  MATH  Google Scholar 

  • Tran, L. T., Roussas, G. G., Yakowitz, S. and Truong Van, B. (1996). Fixed-design regression for linear time series,The Annals of Statistics,24, 975–991.

    Article  MathSciNet  MATH  Google Scholar 

  • Truong, Y. K. (1989). Asymptotic properties of kernel estimators based on local medians,The Annals of Statistics,17, 606–617.

    MathSciNet  MATH  Google Scholar 

  • Truong, Y. K. (1991). Nonparametric curve estimation with time series errors,Journal of Statistical Planning and Inference,28, 167–183.

    Article  MathSciNet  MATH  Google Scholar 

  • Troung, Y. K. (1992a). Robust nonparametric regression in time series,Journal of Multivariate Analysis,41, 163–177.

    Article  MathSciNet  Google Scholar 

  • Truong, Y. K. (1992b). A nonparametric approach for time series analysis.IMA Volumes in Mathematics and Its Applications,45, 371–386.

    MathSciNet  Google Scholar 

  • Truong, Y. K. and Stone, C. J. (1992). Nonparametric function estimation involving time series,The Annals of Statistics,20, 77–97.

    MathSciNet  MATH  Google Scholar 

  • Tsybakov, A. B. (1986). Robust reconstruction of functions by local-approximation method,Problems of Information Transmission,22, 133–146.

    MATH  Google Scholar 

  • Velleman, P. F. (1980). Definition and comparison of robust nonlinear data smoothing algorithms,Journal of the American Statistical Association,75, 609–615.

    Article  MathSciNet  MATH  Google Scholar 

  • Yao, Q. and Tong, H. (1996). Asymmetric least squares regression estimation: a nonparametric approach,Journal of Nonparametric Statistics,6, 273–292.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332-0160, Atlanta, GA, USA

    Liang Peng

  2. Department of Statistics, London School of Economics, WC2A 2AE, London, UK

    Qiwei Yao

Authors
  1. Liang Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiwei Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Peng, L., Yao, Q. Nonparametric regression under dependent errors with infinite variance. Ann Inst Stat Math 56, 73–86 (2004). https://doi.org/10.1007/BF02530525

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02530525

Key words and phrases

Search

Navigation