Closed likelihood-ratio testing procedures to assess similarity of covariance matrices

The study of similarity between k covariance matrices Σh, referred to as k groups, under the assumption of multivariate normality is extended. Our analysis is based on the reparameterization Σh = λhΓh∆hΓ′h where λh, ∆h, and Γh specify volume, shape, and orientation of the density con- tours in each group, respectively. By allowing each of these quantities to be equal or variable between groups, one obtains eight configurations – in which homoscedasticity and heteroscedasticity represent the limit cases – characterized by a different degree of similarity. Due to its desir- able properties in the multiple testing framework, we introduce an easily implementable closed testing procedure allowing for a choice between the eight configurations. Likelihood-ratio tests are used as local tests in the procedure for their optimality properties. The proposed approach discloses a richer information on the data underlying structure than the classical existing methods, the most common one being only based on homo/heteroscedasticity. At the same time, it allows a more parsimonious parameterization, whenever the constrained model is appropriate to describe the real data. The new inferential methodology is finally applied to some well-known data sets, chosen in the multivariate literature, in order to exemplify its use. Our proposal has been also compared with some well-known likelihood-based information criteria