This review is about verifying and generalizing the supremum test statistic developed by Balakrishnan et al. Exhaustive simulation studies are conducted for various dimensions to determine the effect, in terms of empirical size, of the supremum test statistic developed by Balakrishnan et al. to test multivariate skew-normality. Monte Carlo simulation studies indicate that the Type-I error of the supremum test can be controlled reasonably well for various dimensions for given nominal significance levels 0.05 and 0.01. Cut-off values are provided for the number of samples required to attain the nominal significance levels 0.05 and 0.01. Some new and relevant information of the supremum test statistic are reported here.
We generalize the theory of linear models for doubly multivariate data from matrixvariate normally distributed errors to matrix-variate skew normally distributed errors. In addition, we assume that the covariance matrix defining the locationscale matrix-variate skew normal distribution has block compound symmetry structure. We derive the maximum likelihood estimators of the model's parameters; the Fisher information matrix for the direct, working, and centered parametrizations; Rao's score tests and likelihood ratio tests for model building tests of hypotheses; and a hypothesis test for the centered intercept. A profiling argument is used to reduce the dimensionality of the optimization method used to obtain the maximum likelihood estimators and a comprehensive discussion of initial values is provided. Finally, we provide a real-world example to illuminate these derivations and apply a goodness-of-fit test to validate the distributional assumption.
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