Abstract
In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Durbin J (1953). A note on regression when there is extraneous information about one of the coefficients. J Am Stat Assoc 48: 799–808
Farebrother RW (1976). Further results on the mean square error of ridge regression. J R Stat Soc B 38: 248–250
Hoerl AE and Kennard RW (1970). Ridge regression: biased estimation for non-orthogonal problems. Technometrics 12: 55–67
Hubert MH and Wijekoon P (2006). Improvement of the Liu estimator in linear regression model. Stat Pap 47: 471–479
Kaciranlar S, Sakallioglu S, Akdeniz F, Styan GPH and Werner HJ (1999). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland Cement. Sankhya Indian J Stat 61(B): 443–459
Liu K (1993). A new class of biased estimate in linear regression. Commmun Stat Theory Methods 22: 393–402
Massy WF (1965). Principal components regression in exploratory statistical research. JASA 60: 234–266
Rao CR and Toutenburg H (1995). Linear models: least squares and alternatives. Springer, New York
Stein C (1956) Inadmissibility of the usual estimator for mean of multivariate normal distribution. In: Neyman J (ed) Proceedings of the third Berkley symposium on mathematical and statistics probability, vol 1, pp 197–206
Theil H (1963). On the use of incomplete prior information in regression analysis. J Am Stat Assoc 58: 401–414
Theil H and Goldberger AS (1961). On pure and mixed statistical estimation in economics. Int Econ Rev 2: 65–78
Trenkler G and Toutenburg H (1990). Mean squared error matrix comparisons between biased estimators: an overview of recent results. Stat Pap 31: 165–179
Wang SG (2006). Matrix inequalities, 2nd edn. Chinese Science press, Beijing
Woods H, Steinour HH and Starke HR (1932). Effect of composition of Portland cement on heat evolved during hardening. Ind Eng Chem 24: 1207–1241
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, H., Xu, J. An alternative stochastic restricted Liu estimator in linear regression. Stat Papers 50, 639–647 (2009). https://doi.org/10.1007/s00362-007-0102-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-007-0102-3