The sphericity versus equivariance-equicorrelation test

The sphericity and equivariance-equicorrelation tests are two important tests in Multivariate Analysis used, for example, to test assumptions on the structure of covariance matrices which are required in different areas of statistics, for instance in Analysis of Variance and Principal Component Analysis. In this work we combine both tests in a single test, assuming, for the covariance matrix, in the null hypothesis a sphericity structure and in the alternative hypothesis an equivariance-equicorrelation structure. We derive the likelihood ratio test statistic, the expression of its h-th null moment and the expression of the characteristic function of its logarithm, and we show that the exact distribution of the likelihood ratio test statistic is the same as the distribution of the product of independent Beta random variables. Asymptotic approximations are developed in terms of mixtures of Gamma distributions, and numerical studies to assess their quality are developed.