The locally best invariant test for testing sphericity in a multivariate normal distribution is obtained. It is different from the likelihood ratio criterion. Some other known results are remarked. The limiting distributions of the test statistics are investigated. In particular Pitman efficiency of the likelihood ratio test with respect to the locally best invariant test is shown to be one for normal alternatives.
Nariaki Sugiura. "Locally Best Invariant Test for Sphericity and the Limiting Distributions." Ann. Math. Statist. 43 (4) 1312 - 1316, August, 1972. https://doi.org/10.1214/aoms/1177692481