On the Distribution of Hotelling's T² Statistic Based on the Successive Differences Covariance Matrix Estimator

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 accu- rate approximate distribution for calculating the upper control limit for individual observations in a Phase I analysis.