Federal University of Pernambuco Centre for Natural and Exact Sciences Graduate Program in Statistics Modelling and Inference in Integer-valued Time Series

Time series of counts have been the focus of attention due to their importance in several areas of knowledge. The usual stochastic processes assume continuous marginal distributions and therefore are not suitable for modeling series of counts. Thus arises the need to investigate methodologies for time series with discrete marginal distributions. In particular, the study of new models and the behavior of corrected estimators of the integer-valued processes motivates an important research area with practical applications and is the main objective of this work. This work is divided into four independent chapters, all in the context of integer-valued time series. In the second chapter, we introduce a first order non-negative integer-valued autoregressive process with power series innovations. The main properties of the model are derived such as the mean, variance and autocorrelation function. The Yule-Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are obtained. An extensive Monte Carlo experiment is conducted to evaluate the performance of these estimators in finite samples. Applications to two real data sets are given to show the flexibility and potentiality of the new model. In the second third, we consider the first-order Poisson autoregressive model, which may be suitable in situations where the time series data are non-negative integer-valued. We derive the second order bias of the squared difference estimator for one of the parameters and use it to define a bias-adjusted estimator. The behavior of a modified conditional least squares estimator is also studied. Further, we assess the asymptotic properties of the estimators here discussed. We present numerical evidence, based on Monte Carlo simulation studies, that the proposed bias-adjusted estimator outperforms all other estimators in small samples. We also present an application to a real data set. In the fourth chapter, methods based on ranks for estimating the parameters of the Poisson INAR(1) model in the presence of additive outliers are proposed. The effects of additive outliers on the parameters estimates of the integer-valued time series are examined. Some numerical results of the estimators are presented with a discussion of the obtained results. We illustrate the usefulness of the proposed methods by means of an application to a real data set. The results presented motivate the use of the proposed methodology in practical situations in which a Poisson INAR(1) process contains additive outliers. In the fifth chapter, we introduce a stationary first-order integer-valued autoregressive process with geometric-Poisson marginals. The new process allows negative values for the series. Several properties of the process are established. The unknown parameters of the model are estimated by using the Yule-Walker method and their asymptotic properties are considered. Some numerical results of the estimators are presented with a discussion of the obtained results. The flexibility of the new model is illustrated with an application to a real data set.