On Hypothesis Tests for Covariance Matrices Under Multivariate Normality

Abstract

In this paper we proposed a new statistical test for testing the covariance matrix in one population under multivariate normal assumption. In general, the proposed and the likelihood-ratio tests resulted in larger values of estimated powers than VMAX for bivariate and trivariate cases. VMAX was not sensitive to general changes in the covariance (correlation) structure. The advantage of the new test is that it is based on the comparison of all elements of the postulated covariance matrix under the null hypothesis with their respective maximum likelihood sample estimates and therefore, it does not restrict the information of the covariance matrix into a scalar number such as the determinant or trace, for example. Due to the fact that it is based on the maximum likelihood estimates and the Fisher information matrix, it can be used for data coming from distribution other than the multivariate normal.

A New Goodness of Fit Test for Multivariate Normality and Comparative Simulation Study

“…Based on the outcome of normality verification, one can choose suitable analysis methods (parametric or non-parametric) for further investigation. From the end of the 20th century to the present day, multivariate tests for testing the goodness of fit hypothesis have been developed by a number of authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Some of the most popular and commonly used multivariate tests are Chi-Square [8], Cramer von Mises [2], Anderson-Darling [2], and Royston [3].…”

A note on the test for the covariance matrix under normality

“…For this reason, several modifications with high dimensional cases have been reported (Bai et al, 2009;Cai and Ma, 2013;Gupta and Bodnar, 2014) or applications of the bootstrap method, which is a resampling method, have been applied (Beran and Srivastava, 1985). Pinto and Mingoti (2015) also performed a comparison study for the asymptotic LR test with the VMAX proposed by Costa and Machado (2008).…”