# The Likelihood Ratio Test for Equality of Mean Vectors with Compound Symmetric Covariance Matrices

## Abstract

The author derives the likelihood ratio test statistic for the equality of mean vectors when the covariance matrices are assumed to have a compound symmetric structure. Its exact distribution is then expressed in terms of a product of independent Beta random variables and it is shown that for some particular cases it is possible to obtain very manageable finite form expressions for the probability density and cumulative distribution functions for this distribution. For the other cases, given the intractability of the expressions for the exact distribution, very sharp near-exact distributions are developed. Numerical studies show the extreme good performance of these near-exact distributions.

## Keywords

Beta distributions Exact distribution Likelihood ratio statistic Near-exact distributions## Notes

### Acknowledgements

Research supported by FCT–Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology), project UID/MAT/00297/2013, through Centro de Matemática e Aplicações (CMA/FCT-UNL).

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