Linear latent variable models and covariance structures

The asymptotic distribution of the maximum likelihood estimator derived under normality is shown to be valid generally if the different latent variables are independent (not just uncorrelated) and tests of the covariance structure are also asymPTotically robust.

Abstract

The observed vector is taken as a linear combination of latent (unobservable) vector variables. The loading matrices are functions of a vector parameter; the covariance matrix of one latent vector may depend on another vector parameter and the covariance matrices of the other latent vectors are unrestricted positive definite matrices; and the different latent vectors are uncorrelated. The covariance matrix of the observed vector is a function of the two parameter vectors and the covariance matrices of the latent vectors. The maximum likelihood estimator when the latent variables are normally distributed is characterized. The asymptotic distribution of the maximum likelihood estimator derived under normality is shown to be valid generally if the different latent variables are independent (not just uncorrelated). One latent vector is only required to have a sample covariance matrix that has a probability limit. Tests of the covariance structure are also asymptotically robust.