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The joint distribution of the sum of the rth powers (r = 1,..., p-1) and the product of all the latent roots of a p × p Wishart matrix is obtained and used to derive the null distributions of the likelihood ratio test criterion and the locally most powerful invariant test criterion for detecting deviations of the variances and covariances of a p-variate normal distribution from proportionality to specified numbers. The sphericity of the distribution is a special case. Explicit expressions are given for the null distributions in the trivariate case. In the bivariate case the two test criteria coincide and their null distribution has been known. The distribution of the locally most powerful test criterion being complicated for values of p larger than three, some approximations are fitted by the method of moments and compared.
Biometrika © 1972 Biometrika Trust