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779-786) and Xu (1992, M.S. thesis, University of Texas-EI Paso, Mechanical and Industrial Engineering Department, unpublished) using the Monte Carlo experiments for some quite different situations.

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.