Statistics Seminars: Likelihood ratio tests for elaborate structures of covariance matrices
8 May 2017 13:30 in CM221
The analysis and choice of the covariance matrix structure is a key topic in many areas of applied statistics
and modern literature shows that it has become very important to be able to test elaborate structures of the
covariance matrices. However, due to the complicated expressions of the exact distributions of the likelihood
ratio test (LRT) statistics involved, these testing procedures are often not performed or performed using
approximations for the distributions of the LRT statistics which may not guarantee the required accuracy of
the results, mainly in extreme cases such as the ones with small samples and/or large number of variables.
We will show (i) how it is possible to develop LRTs to test elaborate structures of one or several covariance
matrices, and (ii) how may be developed precise and simple near-exact approximations for the distribution
of these LRT statistics. This will be achieved by using expansions for the ratio of two gamma functions
and the similarities exhibited by the distributions of the LRT statistics used to test the equality of several
covariance matrices, the independence of several groups of variables, and the sphericity, compound symmetry
and circular patterns. Examples and numerical studies are presented to illustrate these results and to show
the quality and properties of these approximations.
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