Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model

Affiliation(s)

Department of Mathematics & Statistics, Faculty of Science, University of Jaffna, Jaffna, Sri Lanka;Postgraduate Institute of Science, University of Peradeniya, Sri Lanka.

Department of Statistics & Computer Science, Faculty of Science, University of Peradeniya, Peradeniya, Sri Lanka.

Department of Mathematics & Statistics, Faculty of Science, University of Jaffna, Jaffna, Sri Lanka;Postgraduate Institute of Science, University of Peradeniya, Sri Lanka.

Department of Statistics & Computer Science, Faculty of Science, University of Peradeniya, Peradeniya, Sri Lanka.

ABSTRACT

Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test estimation procedure is adopted to select a suitable estimator. In this paper, two ridge estimators, the Stochastic Restricted Liu Estimator and Liu Estimator are combined to define a new preliminary test estimator, namely the Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE). The stochastic properties of the proposed estimator are derived, and the performance of PTSRLE is compared with SRLE in the sense of mean square error matrix (MSEM) and scalar mean square error (SMSE) for the two cases in which the stochastic restrictions are correct and not correct. Moreover the SMSE of PTSRLE based on Wald (WA), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are derived, and the performance of PTSRLE is compared using WA, LR and LM tests as a function of the shrinkage parameter d with respect to the SMSE. Finally a numerical example is given to illustrate some of the theoretical findings.

Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test estimation procedure is adopted to select a suitable estimator. In this paper, two ridge estimators, the Stochastic Restricted Liu Estimator and Liu Estimator are combined to define a new preliminary test estimator, namely the Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE). The stochastic properties of the proposed estimator are derived, and the performance of PTSRLE is compared with SRLE in the sense of mean square error matrix (MSEM) and scalar mean square error (SMSE) for the two cases in which the stochastic restrictions are correct and not correct. Moreover the SMSE of PTSRLE based on Wald (WA), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are derived, and the performance of PTSRLE is compared using WA, LR and LM tests as a function of the shrinkage parameter d with respect to the SMSE. Finally a numerical example is given to illustrate some of the theoretical findings.

Cite this paper

S. Arumairajan and P. Wijekoon, "Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model,"*Open Journal of Statistics*, Vol. 3 No. 4, 2013, pp. 283-292. doi: 10.4236/ojs.2013.34033.

S. Arumairajan and P. Wijekoon, "Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model,"

References

[1] A. E. Hoerl and R. W. Kennard, “Ridge Regression: Biased Estimation for Nonorthogonal Problems,” Technometrics, Vol. 12, No. 1, 1970, pp. 55-67. doi:10.1080/00401706.1970.10488634

[2] N. Sarkar, “A New Estimator Combining the Ridge Regression and the Restricted Least Squares Methods of Estimation,” Communication in Statistics—Theory Methods, Vol. 21, No. 7, 1992, pp. 1987-2000.

[3] K. Liu, “A New Class of Biased Estimate in Linear Regression,” Communications in Statistics—Theory and Methods, Vol. 22, No. 2, 1993, pp. 393-402. doi:10.1080/03610929308831027

[4] S, Kaciranlar, G. P. H. Styan and H. J. Werner, “A New Biased Estimator in Linear Regression and a Detailed Analysis of the Widely Analyzed Dataset on Portland Cement,” The Indian Journal of Statistics, Vol. 61, No. B3, 1999, pp. 443-459.

[5] M. H. Hubert and P. Wijekoon, “Improvement of the Liu Estimator in Linear Regression Model,” Journal of Statistical Papers, Vol. 47, No. 3, 2006, pp. 471-479. doi:10.1007/s00362-006-0300-4

[6] T. A. Bancroft, “On Biases in Estimation Due to Use of Preliminary Tests of Significance,” Annals of Mathematical Statistics, Vol. 15, No. 2, 1944, pp. 190–204. doi:10.1214/aoms/1177731284

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

[8] P. Wijekoon, “Mixed Estimation and Preliminary Test Estimation in the Linear Regression Model,” PhD Thesis, University of Dortmund, Dortmund, 1990.

[9] A. K. Md. E. Saleh and B. M. G. Kibria, “Performance of Some New Preliminary Test Ridge Regression Estimators and Their Properties,” Communications in Statistics— Theory and Methods, Vol. 22, No. 10, 1993, pp. 27472764.

[10] B. M. G. Kibria and A. K. Md. E. Saleh, “Effect of W, LR and LM Tests on the Performance of Preliminary Test Ridge Regression Estimators,” Journal of the Japan Statistical Society, Vol. 33, No. 1, 2003, pp, 119136.

[11] A. Wald, “Tests of Statistical Hypothesis Concerning Several Parameters When the Number of Observation Is Large,” Transaction of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3

[12] J. Atchison, J. and D. Silvey, “Maximum Likelihood Estimation of Parameters Subject to Restraints,” Annals of Mathematical Statistics, Vol. 29, No. 3, 1958, pp. 813828.doi:10.1214/aoms/1177706538

[13] C. R. Rao, “Large Sample Tests of Statistical Hypothesis Concerning Several Parameters,” Proceeding of the Cambridge Philosophical Society, Vol. 44, No. 1, 1947, pp. 50-57.

[14] H. Yang and J. W. Xu, “Preliminary Test Liu Estimators Based on the Conflicting W, LR and LM Tests in a Regression Model with Multivariate Student-T Error,” Metrica, Vol. 73, No. 3, 2011, pp. 275-292. doi:10.1007/s00184-009-0277-9

[15] H. Theil and A. S. Goldberger, “On Pure and Mixed Estimation in Economics,” International Economic review, Vol. 2, No. 1, 1961, pp. 65-77. doi:10.2307/2525589

[16] J. K. Baksalary, J. K and G. Trenkler, “Nonnegative and Positive Definiteness of Matrices Modified by Two Matrices of Rank One,” Linear Algebra and its Applications, Vol. 151, 1991, pp. 169-184. doi:10.1016/0024-3795(91)90362-Z

[17] C. R. Rao, H. Toutenburg, Shalabh and C. Heumann, “Linear Models: Least Squares and Alternatives” 3rd Edition, Springer, New York, 2006.

[18] G. B. A. Evans and N. E. Savin, “Conflict among the Criteria Revisited; The W, LR and LM Tests,” Econometrica, Vol. 50, No. 3, 1982, pp. 737-748. doi:10.2307/1912611

[19] H. Woods, H. H. Steinour, and H. R. Starke, “Effect of Composition of Portland Cement on Heat Evolved during Hardening,” Industrial and Engineering Chemistry, Vol. 24, No. 11, 1932, pp. 1207-1214. doi:10.1021/ie50275a002

[20] Y. Li and H. Yang, “Two Kinds of Restricted Modified Estimators in Linear Regression Model,” Journal of Applied Statistics, Vol. 38, No. 7, 2011, pp. 1447-1454. doi:10.1080/02664763.2010.505951

[21] Y. Li and H. Yang, “A New Stochastic Mixed Ridge Estimator in Linear Regression,” Journal of Statistical Papers, Vol. 51, No. 2, 2010, pp. 315323.

[1] A. E. Hoerl and R. W. Kennard, “Ridge Regression: Biased Estimation for Nonorthogonal Problems,” Technometrics, Vol. 12, No. 1, 1970, pp. 55-67. doi:10.1080/00401706.1970.10488634

[2] N. Sarkar, “A New Estimator Combining the Ridge Regression and the Restricted Least Squares Methods of Estimation,” Communication in Statistics—Theory Methods, Vol. 21, No. 7, 1992, pp. 1987-2000.

[3] K. Liu, “A New Class of Biased Estimate in Linear Regression,” Communications in Statistics—Theory and Methods, Vol. 22, No. 2, 1993, pp. 393-402. doi:10.1080/03610929308831027

[4] S, Kaciranlar, G. P. H. Styan and H. J. Werner, “A New Biased Estimator in Linear Regression and a Detailed Analysis of the Widely Analyzed Dataset on Portland Cement,” The Indian Journal of Statistics, Vol. 61, No. B3, 1999, pp. 443-459.

[5] M. H. Hubert and P. Wijekoon, “Improvement of the Liu Estimator in Linear Regression Model,” Journal of Statistical Papers, Vol. 47, No. 3, 2006, pp. 471-479. doi:10.1007/s00362-006-0300-4

[6] T. A. Bancroft, “On Biases in Estimation Due to Use of Preliminary Tests of Significance,” Annals of Mathematical Statistics, Vol. 15, No. 2, 1944, pp. 190–204. doi:10.1214/aoms/1177731284

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

[8] P. Wijekoon, “Mixed Estimation and Preliminary Test Estimation in the Linear Regression Model,” PhD Thesis, University of Dortmund, Dortmund, 1990.

[9] A. K. Md. E. Saleh and B. M. G. Kibria, “Performance of Some New Preliminary Test Ridge Regression Estimators and Their Properties,” Communications in Statistics— Theory and Methods, Vol. 22, No. 10, 1993, pp. 27472764.

[10] B. M. G. Kibria and A. K. Md. E. Saleh, “Effect of W, LR and LM Tests on the Performance of Preliminary Test Ridge Regression Estimators,” Journal of the Japan Statistical Society, Vol. 33, No. 1, 2003, pp, 119136.

[11] A. Wald, “Tests of Statistical Hypothesis Concerning Several Parameters When the Number of Observation Is Large,” Transaction of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3

[12] J. Atchison, J. and D. Silvey, “Maximum Likelihood Estimation of Parameters Subject to Restraints,” Annals of Mathematical Statistics, Vol. 29, No. 3, 1958, pp. 813828.doi:10.1214/aoms/1177706538

[13] C. R. Rao, “Large Sample Tests of Statistical Hypothesis Concerning Several Parameters,” Proceeding of the Cambridge Philosophical Society, Vol. 44, No. 1, 1947, pp. 50-57.

[14] H. Yang and J. W. Xu, “Preliminary Test Liu Estimators Based on the Conflicting W, LR and LM Tests in a Regression Model with Multivariate Student-T Error,” Metrica, Vol. 73, No. 3, 2011, pp. 275-292. doi:10.1007/s00184-009-0277-9

[15] H. Theil and A. S. Goldberger, “On Pure and Mixed Estimation in Economics,” International Economic review, Vol. 2, No. 1, 1961, pp. 65-77. doi:10.2307/2525589

[16] J. K. Baksalary, J. K and G. Trenkler, “Nonnegative and Positive Definiteness of Matrices Modified by Two Matrices of Rank One,” Linear Algebra and its Applications, Vol. 151, 1991, pp. 169-184. doi:10.1016/0024-3795(91)90362-Z

[17] C. R. Rao, H. Toutenburg, Shalabh and C. Heumann, “Linear Models: Least Squares and Alternatives” 3rd Edition, Springer, New York, 2006.

[18] G. B. A. Evans and N. E. Savin, “Conflict among the Criteria Revisited; The W, LR and LM Tests,” Econometrica, Vol. 50, No. 3, 1982, pp. 737-748. doi:10.2307/1912611

[19] H. Woods, H. H. Steinour, and H. R. Starke, “Effect of Composition of Portland Cement on Heat Evolved during Hardening,” Industrial and Engineering Chemistry, Vol. 24, No. 11, 1932, pp. 1207-1214. doi:10.1021/ie50275a002

[20] Y. Li and H. Yang, “Two Kinds of Restricted Modified Estimators in Linear Regression Model,” Journal of Applied Statistics, Vol. 38, No. 7, 2011, pp. 1447-1454. doi:10.1080/02664763.2010.505951

[21] Y. Li and H. Yang, “A New Stochastic Mixed Ridge Estimator in Linear Regression,” Journal of Statistical Papers, Vol. 51, No. 2, 2010, pp. 315323.