Tests for Standardized Generalized Variances of Multivariate Normal Populations of Possibly Different Dimensions

摘要

The concept of Standardized Generalized Variances (SGV's) is introduced. Several new problems of multivariate statistical inference are formulated on the basis of these SGV's. It is shown that in addition to providing several new statistical tests, many existing problems of multivariate tests of significance can be regarded as special cases of these formulations and can also be extended to their full generalities. Considering multivariate normal populations with general covariance matrices, likelihood ratio tests are derived for SGV's. The criteria turn out to be elegant multivariate analogues to those for tests of variances in the univariate case. A multivariate F sub max criterion is also presented as an alternative shortcut method for the case of k populations. The null and nonnull distributions of the test criteria are deduced in computable forms (see, e.g., Mathai et al. (1979)) in terms of special functions, e.g., Pincherle's H-function, Meijer's G-function, etc., by exploiting the theory of calculus of residues. Some large sample approximations to these distributions are also proposed. The property of unbiasedness for the modified likelihood ratio tests is established for some of the above test criteria. Finally, applications of the above tests to a wide spectrum of applied research are illustrated by examples taken from the existing literature, e.g., Gnanadesikan (1977), Gnanadesikan and Gupta (1970), etc. (Author)