Minimal complete classes of invariant tests for equality of normal covariance matrices and sphericity

The problem of testing equality of two normal covariance matrices, [Sigma]1 = [Sigma]2 is studied. Two alternative hypotheses, [Sigma]1 [not equal to] [Sigma]2 and [Sigma]1 - [Sigma]2 > 0 are considered. Minimal complete classes among the class of invariant tests are found. The group of transformations leaving the problems invariant is the group of nonsingular matrices. The maximal invariant statistic is the ordered characteristic roots of S1S2-1, where Si, i = 1, 2, are the sample covariance matrices. Several tests based on the largest and smallest roots are proven to be inadmissible. Other tests are examined for admissibility in the class of invariant tests. The problem of testing for sphericity of a normal covariance matrix is also studied. Again a minimal complete class of invariant tests is found. The popular tests are again examined for admissibility and inadmissibility in the class of invariant tests.