  In real data analysis with structural equation modeling, data are unlikely to
be exactly normally distributed. If we ignore the non-normality reality, the
parameter estimates, standard error estimates, and model fit statistics from
normal theory based methods such as maximum likelihood (ML) and normal theory
based generalized least squares estimation (GLS) are unreliable. On the other
hand, the asymptotically distribution free (ADF) estimator does not rely on any
distribution assumption but cannot demonstrate its efficiency advantage with
small and modest sample sizes. The methods which adopt misspecified loss
functions including ridge GLS (RGLS) can provide better estimates and
inferences than the normal theory based methods and the ADF estimator in some
cases. We propose a distributionally-weighted least squares (DLS) estimator,
and expect that it can perform better than the existing generalized least
squares, because it combines normal theory based and ADF based generalized
least squares estimation. Computer simulation results suggest that
model-implied covariance based DLS (DLS_M) provided relatively accurate and
efficient estimates in terms of RMSE. In addition, the empirical standard
errors, the relative biases of standard error estimates, and the Type I error
rates of the Jiang-Yuan rank adjusted model fit test statistic (T_JY) in DLS_M
were competitive with the classical methods including ML, GLS, and RGLS. The
performance of DLS_M depends on its tuning parameter a. We illustrate how to
implement DLS_M and select the optimal a by a bootstrap procedure in a real
data example.
