Higher-order estimation error in structural equation modeling

A general formula of the higher-order asymptotic standard error is derived for the estimators of the parameters in structural equation modeling. The formula can be used for nonnormally distributed data as well as normally distributed ones. For this derivation, the thirdand fourth-order asymptotic central moments of sample variances and covariances are provided for nonnormally and normally distributed cases. The formula requires the partial derivatives of an estimator up to the third order with respect to sample variances and covariances, which are shown for the case of the Wishart maximum likelihood estimator. To see the accuracy of the formula, simulations are performed using the factor/component analysis models. It is numerically shown that some of the added contributions of the higher-order asymptotic standard errors are substantial with small to modest sample sizes.