Change-point detection of mean vector or covariance matrix shifts using multivariate individual observations

A preliminary control chart is given for detecting a shift in the mean vector, the covariance matrix, or both, when multivariate individual observations are available. The data are partitioned after each observation in turn, and the likelihood ratio statistic for a shift is calculated. The control chart is obtained by plotting these statistics after dividing by the expected value under the condition of no shift. This adjustment is done in order to reduce the variation in sensitivity with the location of any shift. Using generalized inverses allows the detection of a shift after as few as two of the observations, or with as few as two remaining observations, or at any intermediate point. Multiple shifts often can be detected by recursive application of the method. When a shift is detected, the plotted statistic is divided into a part due to the shift in the sample mean vector and another part attributable to a shift in the sample covariance matrix. This is done for diagnostic purposes. Using simulation, approximate values are given for the expected values of the plotted statistics and an upper control limit.