线性模型下输入矩阵的最优设计

An Optimum Design of Input Matrix for A Linear Model

线性无偏最小方差估计与最优加权最小二乘估计是线性模型下两种最常用的估计方法 .前者估计性能优于后者 ,但需要被估计量的一、二阶矩信息 .作者深入讨论了这两种估计的关系 ,得到了二者在均方误差意义下等价的充要条件 ,进而找到一种设计输入矩阵的方法 ,使得在先验信息缺乏的条件下 ,仍可利用最优加权最小二乘估计达到与线性无偏最小方差估计一样优越的估计性能 ,避免了获取先验信息的困难 。

The linear unbiased minimum variance estimate and the optimally weighted least squares estimate are two of the most popular estimation methods for a linear model. The former outperforms the latter but it needs priori information on the parameter under estimation.A comprehensive discussion on the relationship between the two estimates is presented and the necessary and sufficient condition, under which the two estimates are identical, is derived. More significantly, the authors know how to design input matrix of the linear system to make the performance of the optimally weighted LS estimation identical to the performance of the linear unbiased minimum variance estimation in the case of being lack of priori information , which avoid the difficulty of getting priori information and retain the optimal estimation.