Improvement of the Liu estimator in linear regression model.

*(English)*Zbl 1125.62055Summary: 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.

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.

##### Keywords:

Ordinary least squares estimator; mixed estimator; Liu estimator; stochastic restricted; mean squared error matrix
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\textit{M. H. Hubert} and \textit{P. Wijekoon}, Stat. Pap. 47, No. 3, 471--479 (2006; Zbl 1125.62055)

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##### References:

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[3] | Liu, K. (1993) A new class of biased estimate in linear regression. Communication in Statistics–Theory and Methods, 22(2), 393–402. · Zbl 0784.62065 |

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[5] | Kaçiranlar, S., Sakallioğlu, S., Akdeniz, F., Styan, G.P.H. and Werner, H.J. (1999) A new biased estimator in linear regression and a detailed analysis of the widely analyzed dataset on Portland Cement. Sankhya: The Indian Journal of Statistics. 61B, 443–459. |

[6] | Thiel, H. and Goldberger, A.S. (1961) On pure and Mixed estimation in Economics. International Economic review, 2, 65–77. |

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