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

• Published in: Statistical methods & applications Vol. 28; no. 4; pp. 593 - 623
• Main Author: Najarzadeh, Dariush
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2019; Springer Nature B.V
• Subjects: Approximation; Chemistry and Earth Sciences; Chi-square test; Computer Science; Covariance matrix; Economics; Equality; Finance; Health Sciences; Humanities; Insurance; Law; Likelihood ratio; Management; Mathematics and Statistics; Medicine; Multivariate analysis; Normal distribution; Normality; Original Paper; Physics; Statistical analysis; Statistical Theory and Methods; Statistics; Statistics for Business; Statistics for Engineering; Statistics for Life Sciences; Statistics for Social Sciences; Standardized generalized variance; Likelihood ratio test; Taylor expansion; Welch–Satterthwaite chi-squared approximation; Alpha adjusted critical level; Bartlett adjustment; Alpha adjusted power
• ISSN: 1618-2510; 1613-981X
• DOI: 10.1007/s10260-019-00456-y

For a
p
-variate normal distribution with covariance matrix
Σ
, the standardized generalized variance (SGV) is defined as the positive
p
th root of
|
Σ
|
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.