On testing hypotheses regarding a class of covariance structures

Let x be ap-component random variable having a multivariate normal distribution with covariance matrix . In this paper, we consider the problem of testing hypotheses of the formH₀: =b₁₁ + +b_mm, whereb_i's are unknown scalars, and _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.