How can I fit a nonstandard mixture (one component is a continuous distribution (skewed to the right) and the other component is a degenerate distribution with mass one at Y=0)? This is what I have tried. In SAS I log-transformed the non-negative variable by first replacing the original zeros by 0.001 (I called it Z1). Then, in Mplus I tried a two-class model where one class should have mean log(0.001) and ZERO variance and the other class a normal distribution N(mu,sigma_square).
You could fix the variance at the small value you chose in order to avoid a zero normal variance. I think you can also as an alternative consider a censored-inflated model with censoring at zero. This is a 2-class model with one class being zero-only people and the other class following a censored-normal distribution. Finally, you might consider a two-part model. All these are described in the User's Guide.
Thanks for your response. I have considered the censored-inflated model with censoring at zero but since the response is strictly non-negative I have a hard time justifying that it is censored at zero (because it is censored at UNKNOWN tau>0). I want the censoring point to be a parameter and hence to be different from the KNOWN "inflation point" 0. In other words, some of the observed zeros are "true zeros" (class of zero-only people) and other are "random zeros" (censored at unknown tau>0). I have also tried the two-part model but I am interested in modeling the marginal distribution of Y not the conditional distribution of Y|Y>0.