Conditions for invariance of the multivariate versions of grubbs’ test and bartlett’s test under a general dependency structure

Summary This paper generalizes a result by Stadje (1984) by deriving conditions for which a general dependency structure for multivariate observations, given in Pavur (1987), yields a positive definite covariance structure. This general dependency structure allows the sample covariance matrix to be distributed as a constant times a Wishart random matrix. It is then demonstrated that the maximum squared-radii test and a test for equal population covariance matrices have null distributions which remain unchanged when the new general dependency structure, rather than the usual independence structure, for the vector observations, is assumed. Moreover, under a general dependency structure for which the population covariance matrices are unequal, it is shown that the distribution of the test statistic for testing equal covariance matrices is identical to the distribution of the same test statistic when the population covariance matrices are equal and the observations are independent.