  Cluster-weighted modeling (CWM) is a mixture approach for modeling the joint
probability of a response variable and a set of explanatory variables. The
parameters are estimated by means of the expectation-maximization algorithm
according to the maximum likelihood approach. Under Gaussian assumptions, we
analyse the complete-data likelihood function of cluster weighted models.
Further, under suitable hypotheses we show that the maximization of the
likelihood function of Gaussian cluster weighted models leads to the same
parameter estimates of finite mixtures of regression and finite mixtures of
regression with concomitant variables. In this sense, the latter ones can be
considered as nested models of Gaussian cluster weighted models.
