Modelos INGARCH log-lineares com inovações Poisson mistas

Abstract

This work proposes a general structure for inference in modelling discrete data sets using a count time series model with INGARCH models with a log-linear structure and mixed Poisson innovations. To this end, a class of probability distributions will be used, the main objective of which is to model count data over time that present a condition of overdispersion. More specifically, the work presents two particular cases: the Inverse Gaussian Poisson log-linear distribution and the Negative Binomial log-linear distribution, which are obtained by considering cases of unobservable data that follow the Inverse Gaussian and Gamma distributions, respectively. The distributions inserted through the mean have as a common point the fact that they are members of the exponential family of distributions. The iterative maximum likelihood method will be used to estimate the model parameters using the EM algorithm. The performance of the estimators will be evaluated through simulation studies using the Monte Carlo method, considering different sample sizes to evaluate the asymptotic behaviour of these estimators.In the section on applying the proposed model to real data sets, three databases were considered for analysis: the first lists the number of hospitalisations due to alcohol abuse in the state of Paraíba, the second evaluates the same problem, but with the data presented for the state of Piauí and, finally, the database consisting of the number of cases of Campylobacter infections in the province of Quebec in Canada was evaluated, thus closing the section on applications to real data. The simulation data was tested using the two proposed extensions and the comparison model called log-linear Poisson proposed by [9], initially taking into account a graphical analysis of the behaviour of the sample, autocorrelation and partial autocorrelation, the study of simulation by means of convergence taking into account the values obtained and the observation of graphs representing a generalised view of the layout of the simulation data. Subsequently, a reflection was made on its effectiveness through information criteria and the mean square error used in the process of evaluating and choosing the best regression model to adjust the data.