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Mean squared error matrix comparisons of some biased estimators in linear regression. (English) Zbl 1028.62054
Summary: Consider the linear regression model $y=X\beta +u$ in the usual notation. In the presence of multicollinearity certain biased estimators like the ordinary ridge regression estimator ${\stackrel{^}{\beta }}_{d}={\left({X}^{\text{'}}X+I\right)}^{-1}{X}^{\text{'}}y$ and the Liu estimator ${\stackrel{^}{\beta }}_{d}={\left({X}^{\text{'}}X+I\right)}^{-1}\left({X}^{\text{'}}y+d\stackrel{^}{\beta }\right)$ introduced by K. Liu [Commun. Stat., Theory Methods 22, 393-402 (1993; Zbl 0784.62065)], or improved ridge and Liu estimators are used to outperform the ordinary least squares estimates in the linear regression model. We compare the (almost unbiased) generalized ridge regression estimator with the (almost unbiased) generalized Liu estimator in the matrix mean square error sense.
##### MSC:
 62J07 Ridge regression; shrinkage estimators 62H12 Multivariate estimation 62J05 Linear regression