Near-exact distributions for a family of generalized sphericity tests
by
Filipe J. Marques & Carlos A. Coelho
Mathematics Department - Faculty of Science and Technology - New University of Lisbon

In this paper a new tecnique is developed to obtain near-exact distributions for the tests of multisample sphericity, block sphericity and multisample block sphericity based on the decomposition of each of the null hypothesis in a number of "partial" hypothesis which are conditionally independent. This way we are able to construct the statistic of each test from the statistics used to test each of the partial hypothesis what allows us to build near-exact distributions for the overall test statistic from the decomposition of the characteristic function of the statistics used to test the partial hypotheses. As partial tests we use the equality of several covariance matrices test, the independence of sets of variates test and the sphericity test. We also develop new asympotitic distributions that are more precise than the existing ones, which are used to show the usefulness of approximations based on moments. Two measures based on the Berry-Esseen inequality are considered to evaluate the quality of the near-exact and asymptotic approximations developed.

Date received: April 6, 2006