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The robust estimation of autoregressive processes by functional least squares

Published online by Cambridge University Press:  14 July 2016

C. R. Heathcote  and
A. H. Welsh
C. R. Heathcote*
Affiliation:
The Australian National University
A. H. Welsh*
Affiliation:
The Australian National University
*
∗ Postal address for both authors: Department of Statistics, Faculty of Economics, The Australian National University, G.P.O. Box 4, Canberra, ACT 2601, Australia.
∗ Postal address for both authors: Department of Statistics, Faculty of Economics, The Australian National University, G.P.O. Box 4, Canberra, ACT 2601, Australia.
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Abstract

The stationary autoregressive model but with a long-tailed error distribution is analysed using the method of functional least squares. A family of estimators indexed by a real parameter is obtained and uniform consistency and weak convergence established. The optimum member of the family is chosen to have minimum variance with respect to the parameter, and the parameter value chosen detects and adjusts for long-tailed error distributions. Results of a simulation are given.

Keywords

AUTOREGRESSIONLONG-TAILED ERRORS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 737 - 753
Copyright
Copyright © Applied Probability Trust 1983 

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References

An, Hong-Zhi and Chen, Zhao-Guo (1982) On convergence of LAD estimates in autoregression with infinite variance. J. Multivariate Analysis 12, 335–345.Google Scholar
Billingsley, P. (1961) The Lindeberg–Lévy theorem for martingales. Proc. Amer. Math. Soc. 12, 788–792.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Birch, J. B. (1977) Small Sample Robustness Properties of Generalised M-estimators Applied to First Order Autoregression. , Department of Biostatistics, University of Washington.Google Scholar
Birch, J. B. and Martin, R. (1981) Confidence intervals for robust estimates of the first order autoregressive parameter. J. Time Series Analysis 2, 205–220.Google Scholar
Campbell, K. (1982) Recursive computation of M-estimates for the parameters of a finite autoregressive process. Ann. Statist. 10, 442–453.CrossRefGoogle Scholar
Chambers, R. L. and Heathcote, C. R. (1981) On the estimation of slope and identification of outliers in linear regression. Biometrika 68, 21–34.Google Scholar
Csörgö, S. (1981) Limit behaviour of the empirical characteristic function. Ann. Prob. 9, 130–144.Google Scholar
Csörgö, S. (1983) The theory of functional least squares. J. Austral. Math. Soc. A 34, 336–355.Google Scholar
Denby, L. and Martin, R. D. (1979) Robust estimation of the first-order autoregressive parameter. J. Amer. Statist. Assoc. 74, 140–146.Google Scholar
Feuerverger, A. and Mureika, R. A. (1977) The empirical characteristic function and its applications. Ann. Statist. 5, 88–97.CrossRefGoogle Scholar
Forsythe, G. E., Malcolm, M. A. and Moler, C. B. (1977) Computer Methods for Mathematical Computation. Prentice Hall, Englewood Cliffs, N.J. Google Scholar
Gross, S. and Steiger, W. L. (1979) Least absolute deviation estimates in autoregression with infinite variance. J. Appl. Prob. 16, 104–116.Google Scholar
Hannan, E. J. and Kanter, M. (1977) Autoregressive processes with infinite variance. J. Appl. Prob. 14, 411–415.Google Scholar
Heathcote, C. R. (1982) Linear regression by functional least squares. J. Appl. Prob. 19A, 225–239.Google Scholar
Kanter, M. and Steiger, W. L. (1974) Regression and autoregression with infinite variance. Adv. Appl. Prob. 6, 768–783.Google Scholar
Marcus, M. B. (1981) Weak convergence of the empirical characteristic function. Ann. Prob. 9, 194–201.Google Scholar
Martin, R. D. (1980) Robust estimation of autoregressive models. In Directions in Time Series, ed. Brillinger, D. R. and Tiao, G. C., Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Tucker, H. G. (1959) A generalization of the Glivenko–Cantelli theorem. Ann. Math. Statist. 30, 828–830; 1267–1268.Google Scholar
Yohai, V. J. and Maronna, R. A. (1977) Asymptotic behaviour of least-squares estimates for autoregressive processes with infinite variances. Ann. Statist. 5, 554–560.Google Scholar