More on the Preliminary Test Stochastic Restricted Liu Estimator in Linear Regression Model

Affiliation(s)

^{1}
Postgraduate Institute of Science, University of Peradeniya, Peradeniya, Sri Lanka.

^{2}
Department of Mathematics and Statistics, Faculty of Science, University of Jaffna, Jaffna, Sri Lanka.

^{3}
Department of Statistics & Computer Science, Faculty of Science, University of Peradeniya, Peradeniya,
Sri Lanka.

ABSTRACT

In this paper we compare recently developed preliminary test estimator called Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE) with Ordinary Least Square Estimator (OLSE) and Mixed Estimator (ME) in the Mean Square Error Matrix (MSEM) sense for the two cases in which the stochastic restrictions are correct and not correct. Finally a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings.

In this paper we compare recently developed preliminary test estimator called Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE) with Ordinary Least Square Estimator (OLSE) and Mixed Estimator (ME) in the Mean Square Error Matrix (MSEM) sense for the two cases in which the stochastic restrictions are correct and not correct. Finally a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings.

KEYWORDS

Multicollinearity, Stochastic Restrictions, Ordinary Least Square Estimator, Mixed Estimator, Preliminary Test Estimator, Mean Square Error Matrix

Multicollinearity, Stochastic Restrictions, Ordinary Least Square Estimator, Mixed Estimator, Preliminary Test Estimator, Mean Square Error Matrix

Cite this paper

Arumairajan, S. , Wijekoon, P. (2015) More on the Preliminary Test Stochastic Restricted Liu Estimator in Linear Regression Model.*Open Journal of Statistics*, **5**, 340-349. doi: 10.4236/ojs.2015.54035.

Arumairajan, S. , Wijekoon, P. (2015) More on the Preliminary Test Stochastic Restricted Liu Estimator in Linear Regression Model.

References

[1] Hoerl, E. and Kennard, W. (1970) Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12, 55-67.

http://dx.doi.org/10.1080/00401706.1970.10488634

[2] Liu, K. (1993) A New Class of Biased Estimate in Linear Regression. Communications in Statistics—Theory and Methods, 22, 393-402.

http://dx.doi.org/10.1080/03610929308831027

[3] Akdeniz, F. and Kaçiranlar, S. (1995) On the almost Unbiased Generalized Liu Estimator and Unbiased Estimation of the Bias and MSE. Communications in Statistics—Theory and Methods, 34, 1789-1797.

http://dx.doi.org/10.1080/03610929508831585

[4] Theil, H. and Goldberger. A.S. (1961) On Pure and Mixed Estimation in Economics. International Economic Review, 2, 65-77.

http://dx.doi.org/10.2307/2525589

[5] Hubert, M.H. and Wijekoon, P. (2006) Improvement of the Liu Estimator in Linear Regression Model. Statistical Papers, 47, 471-479.

http://dx.doi.org/10.1007/s00362-006-0300-4

[6] Bancroft, A. (1944) On Biases in Estimation Due to Use of Preliminary Tests of Significance. Annals of Mathematical Statistics, 15, 190-204.

http://dx.doi.org/10.1214/aoms/1177731284

[7] Judge, G. and Bock, E. (1978) The Statistical Implications of Pre-Test and Stein-Rule Estimators in Econometrics. North Holland, New York.

[8] Wijekoon, P. (1990) Mixed Estimation and Preliminary Test Estimation in the Linear Regression Model. Ph.D. Thesis, University of Dortmund, Dortmund.

[9] Arumairajan, S. and Wijekoon, P. (2013) Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model. Open Journal of Statistics, 3, 283-292.

http://dx.doi.org/10.4236/ojs.2013.34033

[10] Gruber. M.H.J. (1998) Improving Efficiency by Shrinkage: The James-Stein and Ridge Regression Estimators. Dekker, Inc., New York.

[11] Akdeniz, F. and Erol, H. (2003) Mean Squared Error Matrix Comparisons of Some Biased Estimators in Linear Regression. Communications in Statistics—Theory and Methods, 32, 2389-2413.

http://dx.doi.org/10.1081/STA-120025385

[12] Li, Y. and Yang, H. (2010) A New Stochastic Mixed Ridge Estimator in Linear Regression. Statistical Papers, 51, 315-323.

http://dx.doi.org/10.1007/s00362-008-0169-5

[13] Wu, J. and Yang, H. (2013) Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression. Communications in Statistics—Theory and Methods, 42, 3793-3804.

http://dx.doi.org/10.1080/03610926.2011.639004

[14] McDonald, C. and Galarneau, A. (1975) A Monte Carlo Evaluation of some Ridge-Type Estimators. Journal of American Statistical Association, 70, 407-416.

http://dx.doi.org/10.1080/01621459.1975.10479882

[15] Newhouse, J.P. and Oman, S.D. (1971) An Evaluation of Ridge Estimators. Rand Report, No. R-716-Pr, 1-28.

[16] Rao, C.R. and Touterburg, H. (1995) Linear Models, Least Squares and Alternatives. Springer Verlag, Berlin.

http://dx.doi.org/10.1007/978-1-4899-0024-1

[17] Farebrother, R.W. (1976) Further Results on the Mean Square Error of Ridge Regression. Journal of the Royal Statistical Society, 38, 248-250.

[18] Wang, S.G., Wu, M.X. and Jia, Z.Z. (2006) Matrix Inequalities. 2nd Edition, Chinese Science Press, Beijing.

[19] Trenkler, G. and Toutenburg, H. (1990) Mean Square Error Matrix Comparisons between Biased Estimators—An Overview of Recent Results. Statistical Papers, 31, 165-179.

http://dx.doi.org/10.1007/BF02924687

[1] Hoerl, E. and Kennard, W. (1970) Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12, 55-67.

http://dx.doi.org/10.1080/00401706.1970.10488634

[2] Liu, K. (1993) A New Class of Biased Estimate in Linear Regression. Communications in Statistics—Theory and Methods, 22, 393-402.

http://dx.doi.org/10.1080/03610929308831027

[3] Akdeniz, F. and Kaçiranlar, S. (1995) On the almost Unbiased Generalized Liu Estimator and Unbiased Estimation of the Bias and MSE. Communications in Statistics—Theory and Methods, 34, 1789-1797.

http://dx.doi.org/10.1080/03610929508831585

[4] Theil, H. and Goldberger. A.S. (1961) On Pure and Mixed Estimation in Economics. International Economic Review, 2, 65-77.

http://dx.doi.org/10.2307/2525589

[5] Hubert, M.H. and Wijekoon, P. (2006) Improvement of the Liu Estimator in Linear Regression Model. Statistical Papers, 47, 471-479.

http://dx.doi.org/10.1007/s00362-006-0300-4

[6] Bancroft, A. (1944) On Biases in Estimation Due to Use of Preliminary Tests of Significance. Annals of Mathematical Statistics, 15, 190-204.

http://dx.doi.org/10.1214/aoms/1177731284

[7] Judge, G. and Bock, E. (1978) The Statistical Implications of Pre-Test and Stein-Rule Estimators in Econometrics. North Holland, New York.

[8] Wijekoon, P. (1990) Mixed Estimation and Preliminary Test Estimation in the Linear Regression Model. Ph.D. Thesis, University of Dortmund, Dortmund.

[9] Arumairajan, S. and Wijekoon, P. (2013) Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model. Open Journal of Statistics, 3, 283-292.

http://dx.doi.org/10.4236/ojs.2013.34033

[10] Gruber. M.H.J. (1998) Improving Efficiency by Shrinkage: The James-Stein and Ridge Regression Estimators. Dekker, Inc., New York.

[11] Akdeniz, F. and Erol, H. (2003) Mean Squared Error Matrix Comparisons of Some Biased Estimators in Linear Regression. Communications in Statistics—Theory and Methods, 32, 2389-2413.

http://dx.doi.org/10.1081/STA-120025385

[12] Li, Y. and Yang, H. (2010) A New Stochastic Mixed Ridge Estimator in Linear Regression. Statistical Papers, 51, 315-323.

http://dx.doi.org/10.1007/s00362-008-0169-5

[13] Wu, J. and Yang, H. (2013) Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression. Communications in Statistics—Theory and Methods, 42, 3793-3804.

http://dx.doi.org/10.1080/03610926.2011.639004

[14] McDonald, C. and Galarneau, A. (1975) A Monte Carlo Evaluation of some Ridge-Type Estimators. Journal of American Statistical Association, 70, 407-416.

http://dx.doi.org/10.1080/01621459.1975.10479882

[15] Newhouse, J.P. and Oman, S.D. (1971) An Evaluation of Ridge Estimators. Rand Report, No. R-716-Pr, 1-28.

[16] Rao, C.R. and Touterburg, H. (1995) Linear Models, Least Squares and Alternatives. Springer Verlag, Berlin.

http://dx.doi.org/10.1007/978-1-4899-0024-1

[17] Farebrother, R.W. (1976) Further Results on the Mean Square Error of Ridge Regression. Journal of the Royal Statistical Society, 38, 248-250.

[18] Wang, S.G., Wu, M.X. and Jia, Z.Z. (2006) Matrix Inequalities. 2nd Edition, Chinese Science Press, Beijing.

[19] Trenkler, G. and Toutenburg, H. (1990) Mean Square Error Matrix Comparisons between Biased Estimators—An Overview of Recent Results. Statistical Papers, 31, 165-179.

http://dx.doi.org/10.1007/BF02924687