I'm a bit uncertain if I use the right estimator in Mplus in the sense whether it corresponds to the theoretical ones I have studied.
To get a full information maximum likelihood estimates within the context of IRT models (with logit or probit link functions) then one should specify in the Mplus syntax file ESTIMATOR=ML or MLR (and LINK=PROBIT in the case of probit link function). I've read in another thread that it is not really clear the differences between ML and MLR in the case of ordinal data. Did I understand correctly?
In the case of factor analysis with ordinal data using the underlying variable approach there are these GLS limited information estimators. There is also this ML fit function using the polychoric correlation matrix (I think something like Fit_ML= log|Sigma| - log|Polychoric correlation matrix| + trace() - no of parameters to be estimated). Is this last estimator available in Mplus? Under which acronym? What commands should I give?
I think the statement that there are not really clear differences between ML and MLR for categorical variables stems from the fact that MLR is robust to non-normality and that is not an issue for categorical variables. MLR may have advantages over ML in the area of model misspecification.
The Mplus limited information estimator for categorical variables is weighted least squares, WLS, WLSM, and WLSMV.
Mplus can also analyze a polychoric correlation matrix with WLS. This is GLS Incorrect standard errors are obtained with this approach.