Unknown Scalars Research Articles

On testing hypotheses regarding a class of covariance structures.

Let x be ap-component random variable having a multivariate normal distribution with covariance matrix Σ. In this paper, we consider the problem of testing hypotheses of the formH0:Σ =b1Σ1 + … +b m Σ m , whereb i 's are unknown scalars, and Σ i 's are a set of known and simultaneously diagonalizable matrices. This problem has both psychometric and statistical interest, and its basic theory is developed here. Besides, the problem of obtaining likelihood-ratio statistic for testingH0 is studied, and the statistic obtained in a special case.

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 asTΔT′ + Γ where Δ and Γ are both diagonal matrices with unknown elements andT is a known lower triangular matrix, and (b) the covariance matrix Σ*, is expressible asTΔ*T′ + γI 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.