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Improvement of the Liu estimator in linear regression model. (English) Zbl 1125.62055
Summary: In the presence of stochastic prior information, in addition to the sample, H. Thiel and A. S. Goldberger [Int. Econ. Rev. 2, 65–77 (1961)] introduced a mixed estimator \(\widehat{\beta}\) for the parameter vector \(\beta\) in the standard multiple linear regression model \((Y,X\beta,\sigma^2 I)\). Recently, the Liu estimator which is an alternative biased estimator for \(\beta\) has been proposed by K. Liu [Commun. Stat., Theory Methods 22, No. 2, 393–402 (1993; Zbl 0784.62065)].
We introduce another new Liu type biased estimator, called stochastic restricted Liu estimator, \(\widehat{\beta}_{srd}\) for \(\beta\), and discuss its efficiency. Necessary and sufficient conditions for the mean squared error matrix of the stochastic restricted Liu estimator \(\widehat{\beta}_{srd}\) to exceed the mean squared error matrix of the mixed estimator \(\widehat{\beta}_m\) will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the mixed estimator \(\widehat{\beta}_m\) and to the biased estimator introduced by Liu.

MSC:
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
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