On Testing Hypotheses Regarding a Class of Covariance Structures

Let x be a p -component random variable having a multivariate normal distribution with covariance matrix Σ. In this paper, we consider the problem of testing hypotheses of the form H 0 :Σ = b 1 Σ 1 + … + b m Σ m , where b 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 testing H 0 is studied, and the statistic obtained in a special case.