The Nonsingularity of Γ in Covariance Structure Analysis of Nonnormal Data

Covariance structure analysis of nonnormal data is important because in practice all data are nonnormal. When applying covariance structure analysis to nonnormal data, it is generally assumed that the asymptotic covariance matrix Γ for the nonredundant terms in the sample covariance matrix S is nonsingular. It is shown this need not be the case, which raises a question of how restrictive this assumption may be and how difficult it may be to verify it. It is shown that Γ is nonsingular whenever sampling is from a nonsingular distribution, including any distribution defined by a density function. In the discrete case necessary and sufficient conditions are given for the nonsingularity of Γ, and it is shown how to demonstrate Γ is nonsingular with high probability. Thus, the nonsingularity of Γ assumption is mild and one should feel comfortable about making it. These observations also apply to the asymptotic covariance matrix Γ that arises in structural equation modeling.