On the Mean Square Error of Parameter Estimates for Some Biased Estimators
- 1 August 1974
- journal article
- research article
- Published by JSTOR in Technometrics
- Vol. 16 (3) , 461
- https://doi.org/10.2307/1267678
Abstract
Biased estimators of β in the linear statistical model Y = Xβ + e have been studied by several authors. Many solutions are of the form * = (C + X′X)+ X′Y for some compatible symmetric matrix C. In this note, conditions on the matrix C are developed which show when, component by component, the mean square error of the biased estimator is less than the corresponding mean square error, or variance, of the estimators using the normal equations. That is if * i is the ith component of *, and i is the ith component of = (X′X)−1 X′Y, C is determined so that for each i var ( i *) + (β i * – β i )2 < var ( i ). The implication for Hoer1 and Kennard's, Marquardt's, and Mayer and Wilke's biased estimators are discussed. Subsequently, the singular valued decomposition of X is used to explore — *.Keywords
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