Finite mixture models
The mixture model in the previous subsection was a continuous mixture model, as the mixing random variable v was assumed to have continuous distribution. An alternative approach instead uses a discrete representation of unobserved heterogeneity, which generates a class of models called finite mixture models. This class of models is a particular subclass of latent class models.
In empirical work the more commonly used alternative to the continuous mixture is in the class of modified count models discussed in the next section. However, it is more natural to follow up the preceding section with a discussion of finite mixtures. Further, the subclass of modified count models can be viewed as a special case of finite mixtures.
We suppose that the density of у is a linear combination of m different densities, where the jth density is f (у | Xj), j = 1, 2,…, m. Thus an m-component finite mixture is
fj(y|xn) = £njfj(y1 xj), о < n<1 nj =1 (1515)
For example, in a study of the use of medical services with m = 2, the first density may correspond to heavy users of the service and the second to relatively low users, and the fractions of the two types in the populations are n1 and n2(= 1 – щ) respectively.
The goal of the researcher who uses this model is to estimate the unknown parameters Xj, j = 1,…, m. It is easy to develop regression models based on
(15.15) . For example, if NB2 models are used then fj(у | Xj) is the NB2 density (15.13) with parameters p;- = exp(x’Pj) and a;, so Xj = ф;, aj). If the number of
components, m, is given, then under some regularity conditions maximum likelihood estimation of the parameters (nj, Xj), j = 1,…, m, is possible. The details of the estimation methods, less straightforward due to the presence of the mixing parameters nj, is omitted here because of space constraints. See Cameron and Trivedi (1998, ch. 4). It is possible also to probabilistically assign each case to a subpopulation (in the sense that the estimated probability of the case belonging to that subpopulation is the highest) after the model has been estimated.