Proposição de dois testes, sendo um assintótico e outro bootstrap, para comparações entre dois vetores de médias independentes em alta dimensionalidade

Abstract

The inference related to comparisons of mean vectors between two independent populations is of great interest in applied areas, mainly in scenarios where data analyzes with high dimensionality are common. In low-dimensional cases with multivariate Behrens-Fisher problem, there are numerous solutions, but most of the test statistics have an asymptotic distribution. In multivariate procedures there is a problem that arises when the number of variables, p, is greater than or equal to the sample size, n, in this case, it is not possible to use the few existing methods, because they depend on the inverse of the sample covariance matrix which, in this situation (p ≥ n), cannot be obtained, since the coveriance matrix is singular. In most cases, asymptotic tests are very liberal, mainly in small samples and specifically in the multivariate case, when dimensionality is high.The bootstrap method is one of the main computational intensive methods that, among its main advantages, is in the no necessity of knowledge of the population probability distribution. Furthermore, when the conditions assumed for the application of a test are violated, with bootstrap, the problem becomes extremely simple to be circumvented. Based on this, the present work aimed to propose tests of multivariate comparisons between two vectors of independent means, modified Ahmad test (TAM) and its bootstrap version (TB), in high dimensionality, for balanced or unbalanced, nonnormal and normal data under the multivariate Behrens-Fisher problem. The performance of these tests was evaluated and compared with the tests indicated in the literature, being these, Hotelling’s T2, the modified Nel and Merwe test (MNV) proposed by Krishnamoorthy and Yu and the test proposed by Ahmad (TA), using Monte Carlo simulation. Power and type I error rate were considered as evaluative measures. Comparisons were conducted in several scenarios, such as cases of homoscedasticity and heteroscedasticity of covariance matrices, in low and high dimensionality for multivariate distributions normal, t with 7 degrees of freedom and uniform (0, 1), that is, scenarios in which the conditions assumed for the application of most tests are violated. The results showed that the TAM test, in general, was robust, outperforming its competitors in most of the evaluated situations, whereas the test using the bootstrap method was effective in situations of homogeneity of the covariance matrices and in the case of heteroscedasticity, when the matrices are equicorrelated.