chapter  13
19 Pages

Statistical Inference of Covariance Structures

WithAlexander Shapiro

Let X be a p× 1 random vector representing a certain population of interest. Suppose that we hypothesize that the components X1, . . . ,Xp of X are uncorrelated with each other. We can write this as = diag(θ1, . . . , θp), where is the covariance matrix of X and θ = (θ1, . . . , θp)′ is vector of unknown parameters representing respective variances of the components of X . Since these variances should be nonnegative, the parameter vector θ is restricted to the set (nonnegative orthant) Rp+ = {θ : θi ≥ 0, i = 1, . . . , p}. This gives a relatively simple example of a covariance structural model

= (θ), θ ∈ , (13.1)

where (θ) is a symmetric matrix valued function of the parameter vector θ varying in the corresponding parameter space . In order for the matrix (θ) to be nonsingular, we need to enforce constraints θi > 0, i = 1, . . . , p, that is to use the parameter space R

p ++ = {θ : θi > 0, i = 1, . . . , p}.