Statistical Inference of Covariance Structures

¹ Let X be a p ϗ 1 random vector representing a certain population of interest. Suppose that we hypothesize that the components X ₁,…, X_p of X are uncorrelated with each other. We can write this as Σ = diag(θ ₁,…, θ _p), where Σ is the covariance matrix of X and θ = (θ ₁,…, θ _p)′ 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) https://www.w3.org/1998/Math/MathML"> ℝ + p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203813409/28d5761b-f795-4211-a6c9-6a14b869f089/content/inline-math_227_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> = { θ : θ _i ≥ 0, i = 1,…,p}. This gives a relatively simple example of a covariance structural model 13.1 https://www.w3.org/1998/Math/MathML"> Σ = Σ ( θ ) ,   θ ∈ Θ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203813409/28d5761b-f795-4211-a6c9-6a14b869f089/content/math_236_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>