A New Estimation Technique for AR(1) Model with Long-Tailed Symmetric Innovations

  • Ayşen Dener AkkayaEmail author
  • Özlem Türker Bayrak
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


In recent years, it is seen in many time series applications that innovations are non-normal. In this situation, it is known that the least squares (LS) estimators are neither efficient nor robust and maximum likelihood (ML) estimators can only be obtained numerically which might be problematic. The estimation problem is considered newly through different distributions by the use of modified maximum likelihood (MML) estimation technique which assumes the shape parameter to be known. This becomes a drawback in machine data processing where the underlying distribution cannot be determined but assumed to be a member of a broad class of distributions. Therefore, in this study, the shape parameter is assumed to be unknown and the MML technique is combined with Huber’s estimation procedure to estimate the model parameters of autoregressive (AR) models of order 1, named as adaptive modified maximum likelihood (AMML) estimation. After the derivation of the AMML estimators, their efficiency and robustness properties are discussed through simulation study and compared with both MML and LS estimators. Besides, two test statistics for significance of the model are suggested. Both criterion and efficiency robustness properties of the test statistics are discussed, and comparisons with the corresponding MML and LS test statistics are given. Finally, the estimation procedure is generalized to AR(q) models.


Adaptive modified maximum likelihood Autoregressive models Least squares estimators Hypothesis testing Modified maximum likelihood Estimation Efficiency Robustness 


  1. 1.
    Akkaya, A.D., Tiku, M.L.: Estimating parameters in autoregressive models in non-normal situations: asymmetric innovations. Commun. Stat. Theory M 30, 517–536 (2001). Scholar
  2. 2.
    Akkaya, A.D., Tiku, M.L.: Time series AR(1) model for short-tailed distributions. Statistics 39, 117–132 (2005). Scholar
  3. 3.
    Akkaya, A.D., Tiku, M.L.: Autoregressive models in short-tailed symmetric distributions. Statistics 42(3), 207–221 (2008). Scholar
  4. 4.
    Bayrak, O.T., Akkaya, A.D.: Estimating parameters of a multiple autoregressive model by the modified maximum likelihood method. J. Comput. Appl. Math. 233, 1762–1772 (2010). Scholar
  5. 5.
    Dönmez, A.: Adaptive estimation and hypothesis testing methods. PhD thesis, Middle East Technical University (2010)Google Scholar
  6. 6.
    Hamilton, L.C.: Regression with graphics. Brooks/Cole Publishing Company, California (1992)zbMATHGoogle Scholar
  7. 7.
    Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach Based on Influence Functions. Wiley, New York (1986)zbMATHGoogle Scholar
  8. 8.
    Huber, P.J.: Robust Statistics. Wiley, New York (1981)CrossRefGoogle Scholar
  9. 9.
    Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics, vol. 2. Charles Griffin, London (1979)zbMATHGoogle Scholar
  10. 10.
    Tiku, M.L.: Estimating the mean and standard deviation from a censored normal sample. Biometrika 54, 155–165 (1964). Scholar
  11. 11.
    Tiku, M.L., Akkaya, A.D.: Robust estimation and hypothesis testing. New Age International (P) Ltd., New Delhi, India (2004)Google Scholar
  12. 12.
    Tiku, M.L., Sürücü, B.: MMLEs are as good as M-estimators or better. Stat. Prob. Lett. 79(7), 984–989 (2009). Scholar
  13. 13.
    Tiku, M.L., Tan, W.Y., Balakrishnan, N.: Robust Inference. Marcel Dekker, New York (1986)zbMATHGoogle Scholar
  14. 14.
    Tiku, M.L., Wong, W.K., Bian, G.: Estimating parameters in autoregressive models in non-normal situations: symmetric innovations. Commun. Stat. Theory M 28, 315–341 (1999). Scholar
  15. 15.
    Tiku, M.L., Wong, W.K., Vaughan, D.C., Bian, G.: Time series models in non-normal situations. J. Time Ser. Anal. 21, 571–596 (2000). Scholar
  16. 16.
    Ülgen, B.E.: Robust estimation and hypothesis testing in microarray analysis. PhD thesis, Middle East Technical University (2010)Google Scholar
  17. 17.
    Vinod, H.D., Shenton, L.R.: Exact moments for autoregressive and random walk models for a zero or stationary initial value. Economet. Theory 12, 481–499 (1996). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Middle East Technical UniversityAnkaraTurkey
  2. 2.Cankaya UniversityAnkaraTurkey

Personalised recommendations