Asymptotic robustness in multi-sample analysis of multivariate linear relations

Standard methods for the analysis of linear latent variable models often rely on the assumption that the vector of observed variables is normally distributed. This normality assumption (NA) plays a crucial role in assessing optimality of estimates, in computing standard errors, and in designing an asymptotic chi-square goodness-of-fit test. The asymptotic validity of NA inferences when the data deviates from normality has been called asymptotic robustness. In the present paper we extend previous work on asymptotic robustness to a general context of multi-sample analysis of linear latent variable models, with a latent component of the model allowed to be fixed across (hypothetical) sample replications, and with the asymptotic covariance matrix of the sample moments not necessarily finite. We will show that, under certain conditions, the matrix $\Gamma$ of asymptotic variances of the analyzed sample moments can be substituted by a matrix $\Omega$ that is a function only of the cross- product moments of the observed variables. The main advantage of this is that inferences based on $\Omega$ are readily available in standard software for covariance structure analysis, and do not require to compute sample fourth-order moments. An illustration with simulated data in the context of regression with errors in variables will be presented.