Shrinkage estimation of the regression parameters with multivariate normal errors

Document Type : Research Article


Department of Statistics, Ferdowsi University of Mashhad, Mashhad


In the linear model y=XB+e with the errors distributed as normal, we obtain generalized least square (GLS), restricted GLS (RGLS), preliminary test (PT), Stein-type shrinkage (S) and positive-rule shrinkage (PRS) estimators for regression vector parameter \beta when the covariance structure in known. We compare the quadratic risks of the underlying estimators and propose the dominance orders of the five estimators.


[1] Anderson, T.W, An introduction to multivariate statistical analysis, 3rded., John Wiley and Sons, New York, 2003.
[2] Bancroft, T.A., On biases in estimation due to the use of preliminary test of significance, Annals of Math. Stat. 15 (1944), 195-204.
[3] Johnson, N.L. and Kotz, S., Continuous univariate distributions-2, John Wiley, New York, 1970.
[4] Judge, W. and Bock , M.E., The statistical implication of pre-test and Stein-rule estimators in Econometrics, North-Holland, New York, 1978.
[5] Khan, Sh., Improwed estimation of the mean vector for student-t model, Commun. Statist.-Theory Meth. 29(3)(2000), 507-527.
[6] Khan, S., Estimation of parameters of the simple multivariate linear model with Student-t errors, J Statist. Res. 39(2)(2005), 79-94.
[7] Kuan, Ch.M., Statistics: Concepts and Methods, 2nd edition, Hua-Tai, Taipei, 2004.
[8] Ravishanker, N. and Dey, D.K., A first course in linear model theory, Chap-man and Hall/CRC, 2002.
[9] Saleh, A.K.Md.E. and Han, CP., Shrinkage estimation in regression analysis, Estadistica, Vol. 42(1990), 40-63.
[10] Saleh, A.K.Md.E., Theory of Preliminary Test and Stein-type Estimation with Applications, John Wiley, New York, 2006.
[11] Searle, S.R., Matrix Algebra Useful for Statistics, John Wiley, New York, 1982.
[12] Srivastava, M.S. and Sa;eh, A.K.Md.E., Estimation of the mean vactor of a multivariate normal distribution: subspace hypothesis, J. Multivariate Anal-ysis 96(2005), 55-72.
[13] Tabatabaey, S.M.M., Preliminary test approach estimation: regression model with spherically symmetric errors, Ph.D. Thesis, Carleton University, Canada, 1995.