Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 4 (2009), 155 -- 167

ON THE LIU AND ALMOST UNBIASED LIU ESTIMATORS IN THE PRESENCE OF MULTICOLLINEARITY WITH HETEROSCEDASTIC OR CORRELATED ERRORS

M. I. Alheety and B. M. Golam Kibria

Abstract. This paper introduces a new biased estimator, namely, almost unbiased Liu estimator (AULE) of β for the multiple linear regression model with heteroscedastics and/or correlated errors and suffers from the problem of multicollinearity. The properties of the proposed estimator is discussed and the performance over the generalized least squares (GLS) estimator, ordinary ridge regression (ORR) estimator (Trenkler [20]), and Liu estimator (LE) (Kaçiranlar [10]) in terms of matrix mean square error criterion are investigated. The optimal values of d for Liu and almost unbiased Liu estimators have been obtained. Finally, a simulation study has been conducted which indicated that under certain conditions on d, the proposed estimator performed well compared to GLS, ORR and LE estimators.

2000 Mathematics Subject Classification: 62J05; 62F10.
Keywords: Bias, Linear Model; Liu Estimator; MSE; Multicollinearity; OLS; Simulation.

Full text

References

  1. M. I. Alheety and S. D. Gore, A new estimator in multiple linear regression, Model Assisted Statistics and Applications, 3 (3) (2008), 187-200.

  2. F. Akdeniz and S. Kaçiranlar, On the almost unbiased generalized Liu estimator and unbiased estimation of the bias and MSE, Communications in Statistics-Theory and Methods, 24 (1995), 1789-1797. MR1350171. Zbl 0937.62612.

  3. G. M. Bayhan and M. Bayhan, Forecasting using autocorrelated errors and multicollinear predictor variables, Computers and Industrial Engineering, 34 (1998), 413-421.

  4. R. W. Farebrother, Further results on the mean square error of ridge regression, Journal of Royal Statistical Society, B.38 (1976), 284-250. MR0653156(58 #31610). Zbl 0344.62056.

  5. L. Firinguetti, A simulation study of ridge regression estimators with autocorrelated errors, Communications in Statistics - simulation and computation, 18 (2) (1989), 673-702. Zbl 0695.62176.

  6. D. G. Gibbons, A simulation study of some ridge estimators, Journal of the American Statistical Association, 76 (1981), 131139. Zbl 0452.62055.

  7. A. E. Hoerl and R. W. Kennard, Ridge Regression : Biased estimation for non-orthogonal problem, Technometrics, 12 (1970a), 55-67.

  8. A. E. Hoerl and R. W. Kennard, Ridge Regression : Biased estimation for non-orthogonal problem, Technometrics, 12 (1970a), 69-82.

  9. S. Kaçiranlar, S. S. F. Akdeniz, G. P. H. Styan and H. J. Werner, A new biased estimator in linear regression and detailed analysis of the widely-analysed dataset on Portland Cement, Sankhya B, 61 (1999), 443-459.

  10. S. Kaçiranlar, Liu estimator in the general linear regression model, Journal of Applied Statistical Science, 13 (2003), 229-234. MR2038812 (2004k:62170). Zbl 1059.62071.

  11. K. Kadiyala, A class almost unbiased and efficient estimators of regression coefficients, Economics Letter, 16 (1984), 293-296.

  12. B. M. G. Kibria, Performance of some new ridge regression estimators, Communications in Statistics-Simulation and Computation, 32 (2003), 419-435. MR1983342. Zbl 1075.62588.

  13. K. Liu, A new class of biased estimate in linear regression, Communications in Statistics-Theory and Methods, 22 (1993), 393-402. MR1212418.

  14. G. C. Mcdonald and D. I. Galarneau, A monte carlo evaluation of some ridge type estimators, Journal of the American Statistical Association, 20 (1975), 407-416. Zbl 0319.62049.

  15. G. Muniz and B. M. G. Kibria, On Some Ridge Regression Estimators: An Empirical Comparisons, Communications in Statistics - Simulation and Computation, 38 (2009), 621630.

  16. K. Ohtani, On small sample properties of the almost unbiased generalized ridge estimator, Communications in Statistics-Theory and Methods, 15 (1986), 1571-1578. MR0845856. Zbl 0611.62079.

  17. M. R. Özkale, A jackknifed ridge estimator in the linear regression model with heteroscedastic or correlated errors, Statistics and Probability Letters, 78 (18) (2008), 3159-3169. MR2479473. Zbl pre05380098.

  18. M. Quenouille, Notes on bias in estimation, Biometrika, 43 (1956), 3-4, 353-360. MR0081040(18,344f).

  19. C. Stein, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, In : Proceedings of the Third Berkeley Symposium on Mathematics, Statistics and Probability, University of California, Berkeley, I (1956), 197-206. MR0084922(18,948c).

  20. G. Trenkler, On the performance of biased estimators in the linear regression model with correlated or heteroscedastic errors, Journal of Econometrics, 25 (1984), 179-190. MR0748043(86a:62103).

  21. G. Trenkler and H. Toutenburg, Mean squared error matrix comparisons between biased estimators - An overview of recent results, Statistical Papers, 31 (1) (1990), 165-179. MR1124154(92h:62104). Zbl 0703.62066.




Mustafa I. Alheety B. M. Golam Kibria
Department of Mathematics Department of Mathematics and Statistics
Alanbar University, Iraq Florida International University, Miami, FL 33199, USA.
e-mail: alheety@yahoo.com e-mail: kibriag@fiu.edu


http://www.utgjiu.ro/math/sma