圆参数的两步估计(英文)

Two-stage Estimate of the Circle Parameters

在许多工程技术领域经常会遇到用一个圆形去拟合一组观测数据的问题。本文用异方差线性模型提出了圆心和半径的两步估计。利用模型设计阵的特殊结构,建立了最小二乘估计、最佳线性无偏估计和两步估计的协方差阵的很有用的分解式。据此,导出了两步估计的一些重要统计性质。特别,应用Kantorovich不等式获得了两步估计优于最小二乘估计的条件。同时,本文给出了两步估计相当于最佳线性无偏估计相对效率的下界,这个下界表明,既使样本容量不太大,两步估计也有较高的相对效率。最后,本文还证明了观测点在圆周上均匀分布时最小二乘估计的优良性。

The problem of ?tting a circle to a set of observations occurs often in many engineering andtechnology. A two-stage estimate (TSE) of a circle and radius is proposed by using linear model with heteroscedastic variances. Based on special construction of the design matrix, some interesting relations of variance matrices of the least squares estimate (LSE), the best linear unbiased estimate (BLUE) and the TSE are established, and some important properties of the TSE are obtained. In particular, a su?cient condition for the TSE to be superior over the LSE is given. In terms of Kantorovich inequality, a lower bound of a relative e?ciency of the TSE with respect to the BLUE is also obtained. This bound shows, even if the sample size is not large, the TSE also have higher relative e?ciency. Finally, an optimality of the LSE is given when the observations are equally spaced around the circle.