Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

This paper extends the scedasticity comparison among several groups of observations, usually complying with the homoscedastic and the heteroscedastic cases, in order to deal with data sets laying in an intermediate situation. As is well known, homoscedasticity corresponds to equality in orientation, shape and size of the group scatters. Here our attention is focused on two weaker requirements: scatters with the same orientation, but with different shape and size, or scatters with the same shape and size but different orientation. We introduce a multiple testing procedure that takes into account each of the above conditions. This approach discloses a richer information on the data underlying structure than the classical method only based on homo/heteroscedasticity. At the same time, it allows a more parsimonious parametrization, whenever the patterned model is appropriate to describe the real data. The new inferential methodology is then applied to some well-known data sets, chosen in the multivariate literature, to show the real gain in using this more informative approach. Finally, a wide simulation study illustrates and compares the performance of the proposal using data sets with gradual departure from homoscedasticity.