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Statistica Sinica 8(1998), 119-133



Jun Shao and Kwok-Leung Tsui

University of Wisconsin and Hong Kong University of Science & Technology

Abstract: A coordinate measuring machine (CMM) is a computer controlled device that uses a programmable probe to obtain measurements on a part surface. Recently CMMs have become very popular for dimensional measurement in industry due to their flexibility, accuracy, and ease of automation. Despite the advantages offered by CMM's, problems have emerged with their use because tolerance standards require knowledge of the entire surface while a CMM provides only a sample of points on the surface. These problems could be quite challenging, and both practitioners and researchers have shown great interest.

Among these problems, estimating form tolerances for different part features is very important to practitioners. The least squares and minimum zone methods are the most commonly used methods for form tolerance estimation. Dowling et al. (1995) show that these two methods give seriously biased estimates of the part deviation range when the sample size is small. This paper establishes the consistency of these two common estimates. We propose several jackknife estimates that correct the bias of the least squares and minimum zone estimates. Based on a simulation study, it is found that the jackknife estimates effectively reduce the bias of the two common estimates in many situations, and thus reduce the chance of accepting bad parts in tolerance verification. We also show that the jackknife estimates are consistent.

Key words and phrases: Coordinate measuring machine, form tolerance, jackknife, least squares, minimum zone.

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