Modeling Bivariate Mixed-Type Data with the Generalized Linear Exponential Cluster-Weighted Model

In the mixture with random covariates modeling frame, the recently proposed generalized linear Gaussian cluster-weighted model (CWM) allows for flexible clustering and density estimation of a random vector composed by a response variable and by a set of covariates. In each mixture component, while the covariates are assumed to have a realvalued support and are modeled by a Gaussian density, various supports are allowed for the response variable as conceived in the exponential family. For bivariate data, this paper presents the generalized linear exponential CWM. It extends the generalized linear Gaussian CWM by applying an exponential family distribution to the response variable too. This gives the possibility of modeling bivariate data of mixed-type. The natural counterparts, in the frames of mixture models with fixed covariates and latent class models, are also defined and compared with the generalized linear exponential CWM. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criterion (BIC). Artificial and real data are finally considered to exemplify and appreciate the proposed model.