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 + … +bmΣm, wherebi's are unknown scalars, and Σi'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.