Step-Down Analysis for Changes in the Covariance Matrix and Other Parameters

AbstractStep-down diagnostic analysis to infer which means have changed (after detection of a shift in the mean vector) is familiar in multivariate statistical process control. We show how to include in the diagnostic analysis the elements of the covariance matrix and other higher order moments and give practical examples. We use the asymptotic normality of maximum-likelihood estimators to classify the parameters that describe the prior- and post-shift distributions into those for which there is significant evidence of a shift and those for which such evidence is lacking. This diagnostic information can be useful in guiding the search for the cause(s) of the shift. For practical applications, the choice of parameterization may affect the ease of interpretation of the diagnostic results. It may be more informative to use standard deviations and correlation coefficients instead of variances and covariances, and this approach is illustrated in our analysis and examples. We also consider application to distributions other than the multivariate normal and give an example with a multivariate gamma distribution. Although illustrated with independent and identically distributed observations, the proposed methodology is more general.KeywordsDiagnosis of ShiftIndividual ObservationsMultivariate Gamma DistributionRegime ShiftRetrospective AnalysisStatistical Process ControlStructural Change Additional informationNotes on contributorsJoe H. SullivanDr. Sullivan is Professor of Quantitative Analysis and a Member of ASQ. His email address is ymail397-gen2@yahoo.com.Zachary G. StoumbosDr. Stoumbos is Professor and Chair of the Department of Management Science and Information Systems, a Member of the Rutgers Center for Operations Research (RUTCOR), and a Senior Member of ASQ. His email address is stoumbos@andromeda.rutgers.edu.Robert L. MasonDr. Mason is an Institute Analyst and a Fellow of ASQ. His email address is rmason@swri.org.John C. YoungDr. Young is a Professor in the Department of Mathematics, Computer Science, and Statistics. His email address is johncyoung@bellsouth.net.