Saddlepoint approximations for P-values of some tests of covariance matrices

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Abstract
Test statistics for hypotheses about covariance matrices in multivariate analysis of variance have null distributions which are rather intractable to compute. Examples include the modified likelihood ratio statistic for testing (i) homogeneity of covariance matrices across normal populations (Bartlett-Box M-statistic), (ii) homogeneity of normal populations and (iii) simultaneous sphericity of covariances across normal populations. This paper provides two saddlepoint approximations for the null distributions of each of the above mentioned test statistics. These approximations result in extremely accurate p-value computations in all settings including the more difficult unbalanced sample size case. The computations are fast and easily programmed.