I have two strict non-nested models that I would like to compare. However, the normal distribution assumption is violated, so the models were calculated with a robust estimator (MLR).
Using Vuong Test for SEM with MLM;MLR (robust estimator) / under non normality
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Your original post didn't contain a question, but from your comment:
Robust ML estimation is a bit of a misnomer. The point estimates are still obtained via maximum (multivariate-normal) likelihood, and only the SEs and test statistic are corrected for the (estimated) excess kurtosis that impacts them. Vuong's test is a function of casewise (log-)likelihoods, which are still multivariate-normal probability densities when using so-called MLR or MLM. Thus, Vuong's test is not robust to nonnormality. It would be if you were to use a likelihood that allowed for more kurtosis, such as the generalized multivariate skewed t distribution, but I've only seen that implemented in Bayesian programs, in which case there are other more appropriate model-comparison approaches you could use.
You could use Vuong's test if you use standard ML estimation just for the purpose of obtaining the (identical) likelihoods. But I'm not sure how robust Vuong's test is to violations of normality.