Some likelihood ratio tests when a normal covariance matrix has certain reducible linear structures

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Abstract
We first present a review of the literature regarding raducibility of a covarianca matrix that is patterned, eouivalently which has linear structure: ∑ = B1B1 +…+ BmBm where the B's are known matrices and the B's are real numbers. The notion of a pattern is then generalized using the Kronecker product of matrices and some properties derived. Assuming the sampled population is a multivariate normal, we derive likelihood ratio tests for the hypotheses ∑ has linear structure with unknown B's, ∑ has linear structure with given B's, ∑ has linear structure and the B's satisfy certain linear restrictions. Likelihood ratio tests of corresponding Hypotheses when ∑ has a certain Kronecker pattern are seen to be "multivariate" analogues of the previous tests. Some likelihood ratio tests when ∑ has a more general pattern are also derived.