On biased estimation in linear models

Lawrence S. Mayer, Thomas A. Willke

Research output: Contribution to journalArticle

131 Citations (Scopus)

Abstract

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
JournalTechnometrics
Volume15
Issue number3
DOIs
StatePublished - 1973
Externally publishedYes

Fingerprint

Biased Estimation
Linear Model
Estimator
Least Squares Estimator
Ridge
Transform
Minimise
Norm
Euclidean norm
Minimum Variance
Admissibility

Keywords

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

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics
  • Statistics and Probability

Cite this

On biased estimation in linear models. / Mayer, Lawrence S.; Willke, Thomas A.

In: Technometrics, Vol. 15, No. 3, 1973, p. 497-508.

Research output: Contribution to journalArticle

Mayer, LS & Willke, TA 1973, 'On biased estimation in linear models', Technometrics, vol. 15, no. 3, pp. 497-508. https://doi.org/10.1080/00401706.1973.10489076
Mayer, Lawrence S. ; Willke, Thomas A. / On biased estimation in linear models. In: Technometrics. 1973 ; Vol. 15, No. 3. pp. 497-508.
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