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Abstract
In the historical (or retrospective or Phase I) multivariate data analysis, the choice of the estimator for the variance–covariance matrix is crucial to successfully detecting the presence of special causes of variation. For the case of individual multivariate observations, the choice is compounded by the lack of rational subgroups of observations with the same distribution. Other research has shown that the use of the sample covariance matrix, with all of the individual observations pooled, impairs the detection of a sustained step shift in the mean vector. For example, research has shown that, with the use of the sample covariance matrix, the probability of a signal actually decreases below the false-alarm probability with a sustained step shift near the middle of the data and that the signal probability decreases with the size of the shift. An alternative estimator, based on the successive differences of the individual observations, leads to an increasing signal probability as the size of the step shift increases and has been recommended for use in Phase I analysis. However, the exact distribution for the resulting T2 chart statistics has not been determined when the successive differences estimator is used. Three approximate distributions have been proposed in the literature. In this paper, we demonstrate several useful properties of the T2 statistics based on the successive differences estimator and give a more accurate approximate distribution for calculating the upper control limit for individual observations in a Phase I analysis.
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Notes on contributors
James D. Williams
Dr. Williams is a Statistical Leader in the Applied Statistics Laboratory. He is a Member of ASQ. His email address is [email protected].
William H. Woodall
Dr. Woodall is a Professor in the Department of Statistics. He is a Fellow of ASQ. His email address is [email protected].
Jeffrey B. Birch
Dr. Birch is a Professor in the Department of Statistics. His email address is [email protected].
Joe H. Sullivan
Dr. Sullivan is a Professor of Quantitative Analysis. He is a Member of ASQ. His email address is [email protected].
