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Mathematical Problems in Engineering
Volume 2010, Article ID 956907, 30 pages
http://dx.doi.org/10.1155/2010/956907
Research Article

QML Estimators in Linear Regression Models with Functional Coefficient Autoregressive Processes

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received 30 December 2009; Revised 19 March 2010; Accepted 6 April 2010

Academic Editor: Massimo Scalia

Copyright © 2010 Hongchang Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. D. Bai and M. Guo, “A paradox in least-squares estimation of linear regression models,” Statistics & Probability Letters, vol. 42, no. 2, pp. 167–174, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. X. Chen, “Consistency of LS estimates of multiple regression under a lower order moment condition,” Science in China A, vol. 38, no. 12, pp. 1420–1431, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. W. Anderson and J. B. Taylor, “Strong consistency of least squares estimates in normal linear regression,” The Annals of Statistics, vol. 4, no. 4, pp. 788–790, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H. Drygas, “Weak and strong consistency of the least squares estimators in regression models,” Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 34, no. 2, pp. 119–127, 1976. View at Google Scholar · View at MathSciNet
  5. G. González-Rodríguez, A. Blanco, N. Corral, and A. Colubi, “Least squares estimation of linear regression models for convex compact random sets,” Advances in Data Analysis and Classification, vol. 1, no. 1, pp. 67–81, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1986.
  7. X. He, “A local breakdown property of robust tests in linear regression,” Journal of Multivariate Analysis, vol. 38, no. 2, pp. 294–305, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Cui, “On asymptotics of t-type regression estimation in multiple linear model,” Science in China A, vol. 47, no. 4, pp. 628–639, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Durbin, “A note on regression when there is extraneous information about one of the coefficients,” Journal of the American Statistical Association, vol. 48, pp. 799–808, 1953. View at Google Scholar
  10. A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics, vol. 12, no. 1, pp. 55–67, 1970. View at Google Scholar
  11. Y. Li and H. Yang, “A new stochastic mixed ridge estimator in linear regression model,” Statistical Papers. In press. View at Publisher · View at Google Scholar
  12. R. A. Maller, “Asymptotics of regressions with stationary and nonstationary residuals,” Stochastic Processes and Their Applications, vol. 105, no. 1, pp. 33–67, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. P. Pere, “Adjusted estimates and Wald statistics for the AR(1) model with constant,” Journal of Econometrics, vol. 98, no. 2, pp. 335–363, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. A. Fuller, Introduction to Statistical Time Series, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996. View at MathSciNet
  15. G. H. Kwoun and Y. Yajima, “On an autoregressive model with time-dependent coefficients,” Annals of the Institute of Statistical Mathematics, vol. 38, no. 2, pp. 297–309, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case,” Annals of Mathematical Statistics, vol. 29, pp. 1188–1197, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case. II,” Annals of Mathematical Statistics, vol. 30, pp. 831–834, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, NJ, USA, 1994. View at MathSciNet
  19. P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer Series in Statistics, Springer, New York, NY, USA, 1987. View at MathSciNet
  20. K. M. Abadir and A. Lucas, “A comparison of minimum MSE and maximum power for the nearly integrated non-Gaussian model,” Journal of Econometrics, vol. 119, no. 1, pp. 45–71, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  21. F. Carsoule and P. H. Franses, “A note on monitoring time-varying parameters in an autoregression,” Metrika, vol. 57, no. 1, pp. 51–62, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  22. R. Azrak and G. Mélard, “Asymptotic properties of quasi-maximum likelihood estimators for ARMA models with time-dependent coefficients,” Statistical Inference for Stochastic Processes, vol. 9, no. 3, pp. 279–330, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. R. Dahlhaus, “Fitting time series models to nonstationary processes,” The Annals of Statistics, vol. 25, no. 1, pp. 1–37, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. W. Zhang, L. Wei, and Y. Yang, “The superiority of empirical Bayes estimator of parameters in linear model,” Statistics and Probability Letters, vol. 72, no. 1, pp. 43–50, 2005. View at Publisher · View at Google Scholar · View at Scopus
  25. E. J. Hannan, “The asymptotic theory of linear time-series models,” Journal of Applied Probability, vol. 10, pp. 130–145, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. Fox and M. S. Taqqu, “Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series,” The Annals of Statistics, vol. 14, no. 2, pp. 517–532, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. L. Giraitis and D. Surgailis, “A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate,” Probability Theory and Related Fields, vol. 86, no. 1, pp. 87–104, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. Giraitis and H. Koul, “Estimation of the dependence parameter in linear regression with long-range-dependent errors,” Stochastic Processes and Their Applications, vol. 71, no. 2, pp. 207–224, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. H. L. Koul, “Estimation of the dependence parameter in non-linear regression with random designs and long memory errors,” in Proceedings of the Triennial Conference on Nonparametric Inference and Related Fields, Calcutta University, 1997.
  30. H. L. Koul and D. Surgailis, “Asymptotic normality of the Whittle estimator in linear regression models with long memory errors,” Statistical Inference for Stochastic Processes, vol. 3, no. 1-2, pp. 129–147, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. H. L. Koul and K. Mukherjee, “Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors,” Probability Theory and Related Fields, vol. 95, no. 4, pp. 535–553, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  32. T. Shiohama and M. Taniguchi, “Sequential estimation for time series regression models,” Journal of Statistical Planning and Inference, vol. 123, no. 2, pp. 295–312, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York, NY, USA, 2005.
  34. K. N. Berk, “Consistent autoregressive spectral estimates,” The Annals of Statistics, vol. 2, pp. 489–502, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. E. J. Hannan and L. Kavalieris, “Regression, autoregression models,” Journal of Time Series Analysis, vol. 7, no. 1, pp. 27–49, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. Goldenshluger and A. Zeevi, “Nonasymptotic bounds for autoregressive time series modeling,” The Annals of Statistics, vol. 29, no. 2, pp. 417–444, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. E. Liebscher, “Strong convergence of estimators in nonlinear autoregressive models,” Journal of Multivariate Analysis, vol. 84, no. 2, pp. 247–261, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. H.-Z. An, L.-X. Zhu, and R.-Z. Li, “A mixed-type test for linearity in time series,” Journal of Statistical Planning and Inference, vol. 88, no. 2, pp. 339–353, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. R. Elsebach, “Evaluation of forecasts in AR models with outliers,” OR Spectrum, vol. 16, no. 1, pp. 41–45, 1994. View at Publisher · View at Google Scholar · View at Scopus
  40. S. Baran, G. Pap, and M. C. A. van Zuijlen, “Asymptotic inference for unit roots in spatial triangular autoregression,” Acta Applicandae Mathematicae, vol. 96, no. 1–3, pp. 17–42, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. W. Distaso, “Testing for unit root processes in random coefficient autoregressive models,” Journal of Econometrics, vol. 142, no. 1, pp. 581–609, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  42. J. L. Harvill and B. K. Ray, “Functional coefficient autoregressive models for vector time series,” Computational Statistics and Data Analysis, vol. 50, no. 12, pp. 3547–3566, 2006. View at Publisher · View at Google Scholar · View at Scopus
  43. J. S. Simonoff, Smoothing Methods in Statistics, Springer Series in Statistics, Springer, New York, NY, USA, 1996. View at MathSciNet
  44. P. J. Green and B. W. Silverman, Nonparametric Regression and Generalized Linear Models, vol. 58 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, UK, 1994. View at MathSciNet
  45. J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, vol. 66 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, UK, 1996. View at MathSciNet
  46. Y. Fujikoshi and Y. Ochi, “Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process,” Annals of the Institute of Statistical Mathematics, vol. 36, no. 1, pp. 119–128, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. C. R. Rao, Linear Statistical Inference and Its Applications, Wiley Series in Probability and Mathematical Statistic, John Wiley & Sons, New York, NY, USA, 2nd edition, 1973. View at MathSciNet
  48. R. A. Maller, “Quadratic negligibility and the asymptotic normality of operator normed sums,” Journal of Multivariate Analysis, vol. 44, no. 2, pp. 191–219, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application. Probability and Mathematical Statistic, Academic Press, New York, NY, USA, 1980. View at MathSciNet