On biased estimation in linear models

Lawrence S. Mayer, Thomas A. Willke

Research output: Contribution to journalArticlepeer-review

142 Scopus citations


Hoer1 and Kennard introduced a class of biased estimators (ridge estimators) for the parameters in an ill-conditioned linear model. In this paper the ridge estimators are viewed as a subclass of the class of linear transforms of the least squares estimator. An alternative class of estimators, labeled shrunken estimators is considered. It is shown that these estimators satisfy the admissibility condition proposed by Hoer1 and Kennard. In addition, both the ridge estimators and shrunken estimators are derived as minimum norm estimators in the class of linear transforms of the least squares estimators. The former minimizes the Euclidean norm and the latter minimizes the design dependent norm. The class of estimators which are minimum variance linear transforms of the least squares estimator is obtained and the members of this class are shown to be stochastically shrunken estimators. An example is computed to show the behavior of the different estimators.

Original languageEnglish (US)
Pages (from-to)497-508
Number of pages12
Issue number3
StatePublished - Aug 1973


  • Biased estimation
  • Ill-conditioning
  • Least squares
  • Linear models
  • Multicollinearity
  • Regression
  • Ridge estimators
  • Shrunken estimators

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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