Abstract
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.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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). https://doi.org/10.1081/STA-100002095
Akkaya, A.D., Tiku, M.L.: Time series AR(1) model for short-tailed distributions. Statistics 39, 117–132 (2005). https://doi.org/10.1080/02331880512331344036
Akkaya, A.D., Tiku, M.L.: Autoregressive models in short-tailed symmetric distributions. Statistics 42(3), 207–221 (2008). https://doi.org/10.1080/02331880701736663
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). https://doi.org/10.1016/j.cam.2009.09.013
Dönmez, A.: Adaptive estimation and hypothesis testing methods. PhD thesis, Middle East Technical University (2010)
Hamilton, L.C.: Regression with graphics. Brooks/Cole Publishing Company, California (1992)
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach Based on Influence Functions. Wiley, New York (1986)
Huber, P.J.: Robust Statistics. Wiley, New York (1981)
Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics, vol. 2. Charles Griffin, London (1979)
Tiku, M.L.: Estimating the mean and standard deviation from a censored normal sample. Biometrika 54, 155–165 (1964). https://doi.org/10.2307/2333859
Tiku, M.L., Akkaya, A.D.: Robust estimation and hypothesis testing. New Age International (P) Ltd., New Delhi, India (2004)
Tiku, M.L., Sürücü, B.: MMLEs are as good as M-estimators or better. Stat. Prob. Lett. 79(7), 984–989 (2009). https://doi.org/10.1016/j.spl.2008.12.001
Tiku, M.L., Tan, W.Y., Balakrishnan, N.: Robust Inference. Marcel Dekker, New York (1986)
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). https://doi.org/10.1080/03610929908832300
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). https://doi.org/10.1111/1467-9892.00199
Ülgen, B.E.: Robust estimation and hypothesis testing in microarray analysis. PhD thesis, Middle East Technical University (2010)
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). https://doi.org/10.1017/S0266466600006824
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Akkaya, A.D., Bayrak, Ö.T. (2018). A New Estimation Technique for AR(1) Model with Long-Tailed Symmetric Innovations. In: Rojas, I., Pomares, H., Valenzuela, O. (eds) Time Series Analysis and Forecasting. ITISE 2017. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-96944-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-96944-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96943-5
Online ISBN: 978-3-319-96944-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)