Statistical Methods & Applications

, Volume 28, Issue 4, pp 593–623 | Cite as

Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

  • Dariush NajarzadehEmail author
Original Paper


For a p-variate normal distribution with covariance matrix \( {\varvec{\Sigma }}\), the standardized generalized variance (SGV) is defined as the positive pth root of \( |{\varvec{\Sigma }}| \) and used as a measure of variability. Testing equality of the SGVs, for comparing the variability of multivariate normal distributions with different dimensions, is still regarded as matter of interest. The most classical test for this problem is the likelihood ratio test (LRT). In this article, testing equality of the SGVs of k multivariate normal distributions with possibly unequal dimensions is studied. To test this hypothesis, two approximations for the null distribution of the LRT statistic are proposed based on the well known Welch–Satterthwaite and Bartlett adjustment distribution approximation methods. Furthermore, the high-dimensional behavior of these approximated distributions is also investigated. Through a wide simulation study: first, the performance of the proposed tests with the classical LRT is compared in terms of type I error, power, and alpha adjusted equivalents; second, the robustness of the procedures with respect to departures from normality assumption is evaluated. Finally, the proposed methods are illustrated with two real data examples.


Standardized generalized variance Likelihood ratio test Welch–Satterthwaite chi-squared approximation Bartlett adjustment Taylor expansion Alpha adjusted critical level Alpha adjusted power 



We would like to express our sincere thanks to the editor and the two anonymous reviewers for their comments which greatly improved this article. The corresponding author would like to thank the “Iranian National Elites Foundation” for the financial support of this research.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of Mathematical SciencesTabriz UniversityTabrizIran

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