New models for observations that taken values on integers
Paired counts samples. Skew discrete Laplace distribution. Regression models. Non-stationary counting processes. Integer-valued times series.
There are several practical situations in which there is interest in modeling events associated with discrete-valued random variables. Until now, theories that have been constructed and refined to handling observations of this nature have an emphasis on modeling non-negative discrete data. Nevertheless, discrete observations that can assume any value on the set ofintegers Z = {..., -2, -1, 0, 1, 2, ...}, including negative observations, can also be found in different contexts. The aims of this master thesis are to propose a new parameterization for the skewdiscrete Laplace distribution (LDA), in terms of the mean and a dispersion parameter, and then define two new parametric models able of modeling observations that assume values on Z, based on this distribution. The first one consists of a regression model in which it is assumed that the response variable follows the LDA distribution. We consider the maximum likelihoodestimator as an estimation method for the model parameters. We propose diagnostic methods to evaluate the goodness of fit. We performed simulation studies to verify the performance ofthe properties obtained in finite samples size and we apply the model to a set of real data in the area of psychology. The second model consists of an autoregressive integer-valued processwith innovations distributed according to the LDA distribution. Its main properties are presented, two different methods of estimation for the parameters of the model are defined, and it is also considered a regression structure that allows the introduction of covariates that may be associated with observations.