Abstract: The least squares estimation is widely used in linear models. However, in the least squares estimation, the square error may be great, when there are a multiple collinearity between variables. To solve this problem, it is suggested that, instead of an unbiased estimate, a biased estimate is used. As a biased estimate, the generalized ridge estimate is an estimate in a wide use. In many practical problems, aggregated data can be observed. For a linear model with aggregated data, the definition of aggregated Liu estimates is given in this paper. The Liu estimates with respect to two relative efficiencies of the least squares estimation are proposed and the upper bounds for the two relative efficiencies are obtained. This paper also gives the aggregated Liu estimates relative to Peter-Karsten estimates for two relative efficiencies and their upper bounds. The aggregation of generalized ridge estimates was often said to reduce the mean square error, and the stability of estimated parameters was emphasized, ignoring the non-bias effect of the estimated parameters. Aggregated Liu estimates, presented in this paper, with the introduction of new parameters, can not only guarantee the stability of estimated parameters, but also ensure the approximate unbiasedness of the estimated parameters. In this sense, they are better than the aggregated generalized ridge estimates.
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Received: 29 March 2010
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