聚集数据线性模型参数聚集Liu估计的相对效率

 当线性模型中的变量间存在复共线性时，常用有偏估计代替无偏估计。其中广义岭估计是研究较多的一种有偏估计。很多实际问题只能观测到聚集数据。本文给出了聚集数据线性模型聚集Liu估计的定义，提出了聚集Liu估计相对于最小二乘估计的两种相对效率，并得到这两种相对效率的上界；给出了聚集Liu估计相对于Peter-Karsten估计的2种相对效率及其上界。本文提出的聚集Liu估计，既能保证估计参数的稳定性，又能保证估计参数的近似无偏性，从这个意义上说，该估计在某种程度上优于聚集广义岭估计。

Relative Efficiencies of the Aggregated Liu Estimators of the Parameters in a Linear Model with Aggregated Date

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.