Derivation of Likelihood-Ratio Tests for Guttman Quasi-Simplex Covariance Structures

In this paper, maximum-likelihood estimates have been obtained for covariance matrices which have the Guttman quasi-simplex structure under each of the following null hypotheses: (a) The covariance matrix Σ, can be written as TΔT′ + Γ where Δ and Γ are both diagonal matrices with unknown elements and T is a known lower triangular matrix, and (b) the covariance matrix Σ*, is expressible as TΔ*T′ + γ I where γ is an unknown scalar. The linear models from which these covariance structures arise are also stated along with the underlying assumptions. Two likelihood-ratio tests have been constructed, one each for the above null hypotheses, against the alternative hypothesis that the population covariance matrix is simply positive definite and has no particular pattern. A numerical example is provided to illustrate the test procedure. Possible applications of the proposed test are also suggested.