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Title: Likelihood ratio test for the double level compound symmetric structure Authors:  Vusi Bilankulu - University of Pretoria (South Africa) [presenting]
Andriette Bekker - University of Pretoria (South Africa)
Carlos Coelho - NOVA University of Lisbon (Portugal)
Abstract: It is shown how from an adequate decomposition of the null hypothesis of the double level block compound symmetric covariance structure it is easy to obtain the expression for the likelihood ratio test (l.r.t.) statistic to test this hypothesis, as well as the general expression for its moments. From this expression it is then possible to identify the structure of the exact distribution and to obtain the characteristic function of the negative logarithm of the l.r.t. statistic. Then, from an adequate decomposition of this characteristic function it is possible to identify the distribution components that will be left untouched and those that will have to be asymptotically approximated. Based on this identification, near-exact distributions are then built. They are shown to be asymptotic both for increasing sample sizes, as well as for increasing numbers of variables and increasing numbers of groups of variables, while common asymptotic distributions are only asymptotic for sample size. Moreover, these near-exact distributions show very good performances for very small samples, enabling us to obtain very good approximations for situations where being $p$ the number of variables involved and $n$ the sample size, $p/n$ approaches 1 from below.