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。

Comparison is made between the MINQUE and simple estimate of the error variance in the normal linear model under the mean square errors criterion, where the model matrix need not have full rank and the dispersion matrix can be singular. Our results show that any one of both estimates cannot be always superior to the other. Some sufficient criteria for any one of them to be better than the other are established. Some interesting relations between these two estimates are also given.

线性混合模型的未知参数分两类，一类是固定效应，一类是方差分量。提出了固定效应和方差分量的一种新估计，称为谱分解估计，并证明了新估计的一些重要统计性质．新估计的突出特点是，固定效应的估计是具有良好统计性质的线性估计。新方法破应用于经济、金融和机械测量方面的两个重要模型，所得到的估计都具有很好的统计和实用意义。

对于具有广泛应用的含有两个方差分量的线性混合模型，找到了一组简单条件。在这些条件下，证明了固定效应的最小二乘估计和方差分量的方差分析估计同时是最小方差无偏估计；获得了固定效应的精确置信区问和随机效应的方差分量的一致最优无偏检验；得到了随机效应方差的方差分析估计取负值的概率精确表达式。

A coordinate measuring machine (CMM) is a computer-controlled device that uses a probe to obtain measurements on a manufactured part's surface. In the process of collecting, analyzing and interpreting CMM data, many statistical problems arise. One of them is to choose a model describing the relationship between the location and shape parameters of the part and CMM data and representing the effects of the various sources of randomness of these data. This article suggests a linear model for a stratified sampling scheme, which is one of the most commonly discussed in the CMM literature, in fitting a spherical surface. A feasible generalized least-squares estimator of the part's spherical parameter set is given and its property is studied. Our theoretical results indicate that stratified sampling performs better than random sampling. A similar conclusion was also obtained by Caskey et al. (1990, Design Manufacturing Systems Conf. 779-786) and Xu (1992, M.S. thesis, University of Texas-EI Paso, Mechanical and Industrial Engineering Department, unpublished) using the Monte Carlo experiments for some quite different situations.