Mixture models for count data: the ZIP model

~ 4 ~ Preface In the context of Inference and Statistical Models, to describe any phenomenon, many times the basic tools, that we can use, it doesn’t enough to explain the real phenomenon that we try to model through the common statistical model. Therefore, in this particular case, to overcome several specification problems and to overcome some limited use of the basic model it is formulated a special class of models: mixture models. Mixture models have gained an increased popularity in many fields of sciences; the main feature of this class of model is, in fact, that commonly used density functions are employed as building blocks for more complex distributions: this allows for a great flexibility in statistical modeling and makes mixture models adequate to very complicate framework. In this work, I have underscored also, some of the reasons that leads to fit a mixture model, like: the presence of k unobserved subpopulation (see Bruce G. Lindsay (1995) in: “Mixture Models: Theory, Geometry and Applications“), or even when there is no reason to believe that a latent structure affects the data- generating process, a mixture models can be fitted with the aim of exploiting its flexibility, as pointed out in McLachlan and Peel (2000). As stated Lindsay (1995), a mixture model has a dual usefulness: on the one hand it enables to study the distribution of the outcome response (Y) when a covariate (Z) is missing; on the other hand, it makes use of a surrogate measure (Y) to learn about an unobserved variable (Z). Unfortunately, estimating the parameters of a mixture model presents a number of obstacles: first, model identification is not guaranteed; second, estimates are sensitive to the started values used for the optimization algorithm. In this work I shortly reviews some of this problems. Farther, I gutted the EM- algorithm that allows to overcome this estimate’s problems, reporting step by step whole formalization of that iterative process. Finally, I have been studying a very particular mixture model, the Zero Inflated Poisson model (ZIP), which help us when the data have an excess number of zeros. (There are other mixture model, for example, that help us with count variables when the classical Poisson regression model are limited because the data exhibit over or under dispersion. Also in this case we can use ZIP model or Negative Binomial that is another type of mixture model). I also fit ZIP model on real data with statistical software R. The real data that I will use for my ZIP application, will a dataset of 133707 observations on three variables, that I was provided by ANIA-National Association of Insurance Companies, after a very long practice.