Subsections:

The general expression for the finite mixture model fitted with the HPFMM procedure is as follows:

The number of components in the mixture is denoted as k. The mixture probabilities can depend on regressor variables and parameters . By default, the HPFMM procedure models these probabilities using a logit transform if k = 2 and as a generalized logit model if k > 2. The component distributions can also depend on regressor variables in , regression parameters , and possibly scale parameters . Notice that the component distributions are indexed by j since the distributions might belong to different families. For example, in a two-component model, you might model one component as a normal (Gaussian) variable and the second component as a variable with a t distribution with low degrees of freedom to manage overdispersion.

The mixture probabilities satisfy , for all j, and