Closed Likelihood Ratio Testing Procedures to Assess Similarity of Covariance Matrices

In this paper we introduce a multiple testing procedure to assess a common covariance structure between k groups. The new test allows for a choice among eight different patterns arising from the three-term eigen decomposition of the group covariances. It is based on the closed testing principle and adopts local likelihood ratio tests. The approach reveals richer information about the underlying data structure than classical methods, the most common one being only based on homo/heteroscedasticity. At the same time, it provides a more parsimonious parameterization, whenever the constrained model is suitable to describe the real data. The new inferential methodology is then applied to some well-known data sets chosen from the multivariate literature. Finally, simulation results are presented to investigate its performance in different situations representing gradual departures from homoscedasticity and to evaluate the reliability of using the asymptotic χ2 to approximate the actual distribution of the local likelihood ratio test statistics.