Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-25T17:24:41.424Z Has data issue: false hasContentIssue false
  • English
  • Français

M-ESTIMATION FOR A SPATIAL UNILATERAL AUTOREGRESSIVE MODEL WITH INFINITE VARIANCE INNOVATIONS

Published online by Cambridge University Press:  17 March 2010

S.M. Roknossadati  and
M. Zarepour
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the limiting behavior of the M-estimators of parameters for a spatial unilateral autoregressive model with independent and identically distributed innovations in the domain of attraction of a stable law with index α ∈ (0, 2]. Both stationary and unit root models and some extensions are considered. It is also shown that self-normalized M-estimators are asymptotically normal. A numerical example and a simulation study are also given.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 6 , December 2010 , pp. 1663 - 1682
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper is supported by the Natural Sciences and Engineering Research Council of Canada. The first author was also supported by the Central Bank of Iran. The authors also are greatly indebted to two anonymous referees for several helpful comments.

References

REFERENCES

Basu, S. & Reinsel, G.C. (1992a) Regression Models with Spatially Correlated Errors. Technical report, Department of Statistics, University of Wisconsin-Madison.Google Scholar
Basu, S. & Reinsel, G.C. (1992b) A note on properties of spatial Yule-Walker estimators. Journal of Statistical Computation and Simulation 41, 243–255.CrossRefGoogle Scholar
Basu, S. & Reinsel, G.C. (1993) Properties of the spatial unilateral first-order ARMA model. Advances in Applied Probability 25, 631–648.CrossRefGoogle Scholar
Basu, S. & Reinsel, G.C. (1994) Regression models with spatially correlated errors. Journal of the American Statistical Association 89, 88–99.CrossRefGoogle Scholar
Bhattacharyya, B.B., Khalil, T.M., & Richardson, G.D. (1996) Gauss-Newton estimation of parameters for a spatial autoregression model. Statistics and Probability Letters. 28, 173–179.10.1016/0167-7152(95)00114-XCrossRefGoogle Scholar
Bhattacharyya, B.B., Richardson, G.D., & Franklin, L.A. (1997) Asymptotic inference for near unit roots in spatial autoregression. Annals of Statistics 25, 1709–1724.CrossRefGoogle Scholar
Bickel, P.J. & Wichura, M.J. (1971) Convergence criteria for multiparameter stochastic processes and some applications. Annals of Mathematical Statistics 42, 1656–1670.CrossRefGoogle Scholar
Bronars, S.G. & Jansen, D.W. (1987) The geographic distribution of unemployment in the U.S.: A spatial-time series analysis. Journal of Econometrics 36, 251–279.CrossRefGoogle Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.10.1214/aoms/1177693494CrossRefGoogle Scholar
Chellappa, R. (1985) Two-dimensional discrete Gaussian Markov random field models for image processing. In Kanal, L.N. & Rosenfeld, A. (eds.), Progress in Pattern Recognition 2, pp. 79–112. North-Holland.Google Scholar
Cullis, B.R. & Gleeson, A.C. (1991) Spatial analysis of field experiments—An extension to two dimensions. Biometrics 47, 1449–1460.CrossRefGoogle Scholar
Dass, S.C. & Nair, V.N. (2003) Edge detection, spatial smoothing, and image reconstruction with partially observed multivariate data. Journal of the American Statistical Association 98, 77–89.CrossRefGoogle Scholar
Davis, R.A. (1996) Gauss-Newton and M-estimates for ARMA processes with infinite variance. Stochastic Processes and Their Applications 63, 75–95.CrossRefGoogle Scholar
Davis, R.A., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and Their Applications 40, 145–180.10.1016/0304-4149(92)90142-DCrossRefGoogle Scholar
Davis, R. & Resnick, S.I. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Annals of Probability 13, 179–195.CrossRefGoogle Scholar
Davis, R.A. & Mikosch, T. (2008) Extreme value theory for space-time processes with heavy-tailed distributions. Stochastic Processes and Their Applications 118, 560–584.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Application, vol. 2, 2nd ed. Wiley.Google Scholar
Geman, S. & Geman, D. (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–741.10.1109/TPAMI.1984.4767596CrossRefGoogle ScholarPubMed
Huber, P.J. (1981) Robust Statistics, Wiley Series in Probability and Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Jain, A.K. (1981) Advances in mathematical models for image processing. Proceedings of IEEE 69, 502–528.CrossRefGoogle Scholar
Kallenberg, O. (1983) Random Measures. Akademie-Verlag.CrossRefGoogle Scholar
Kempton, R.A. & Howes, C.W. (1981) The use of neighbouring plot values in the analysis of variety trials. Applied Statistics 30, 59–70.CrossRefGoogle Scholar
Knight, K. (1989) Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261–278.CrossRefGoogle Scholar
Le Page, R., Woodroofe, M., & Zinn, J. (1981) Convergence to a stable distribution via order statistics. Annals of Probability 9, 624–632.Google Scholar
Martin, R.J. (1979) A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66, 209–217.CrossRefGoogle Scholar
Martin, R.J. (1990) The use of time-series models and methods in the analysis of agricultural field trials. Communications in Statistical Theory and Methods 19, 55–81.CrossRefGoogle Scholar
McLeish, D.L. (1974) Dependent central limit theorems and invariance principles. Annnals of Probability 2, 620–628.Google Scholar
Resnick, S.I. & Greenwood, P. (1979) A bivariate stable characterization and domain of attraction. Journal of Multivariate Analysis 9, 206–221.CrossRefGoogle Scholar
Resnick, S.I. (1986) Point processes, regular variation and weak convergence. Advances in Applied Probability 18, 66–138.CrossRefGoogle Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag.CrossRefGoogle Scholar
Resnick, S.I. (2007) Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer-Verlag.Google Scholar
Samorodnitsky, G. & Taqqu, M. (1994) Stable Non-Gaussian Random Processes. Chapman and Hall.Google Scholar
Tjøstheim, D. (1978) Statistical spatial series modelling. Advances in Applied Probability 10, 130–154.CrossRefGoogle Scholar
Tjøstheim, D. (1981) Autoregressive modeling and spectral analysis of array data in the plane. IEEE Transactions on Geoscience and Remote Sensing 19, 15–24.CrossRefGoogle Scholar
Tjøstheim, D. (1983) Statistical spatial series modelling. II. Some further results on unilateral lattice processes. Advances in Applied Probability 15, 562–584.CrossRefGoogle Scholar
Van de Geer, S. (2000) Empirical Processes in M-Estimation, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.Google Scholar
Walsh, J.B. (1986) Martingales with a multidimensional parameter and stochastic integrals in the plane. In Lectures in Probability and Statistics, Lecture Notes in Mathematics 1215, pp. 329–491. Springer-Verlag.CrossRefGoogle Scholar
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–449.CrossRefGoogle Scholar
Zarepour, M. & Roknossadati, S.M. (2008) Multivariate autoregression of order one with infinite variance innovations. Econometric Theory 24, 677–695.CrossRefGoogle Scholar