O processo de Poisson estendido e aplicações - Salasar, Luis Ernesto Bueno (Universidade Federal de São Carlos, 2007)
Abstract In this dissertation we will study how extended Poisson process can be applied to construct discrete probabilistic models. An Extended Poisson Process is a continuous time stochastic process with the state space being the natural numbers, it is obtained as a generalization of homogeneous Poisson process where transition rates depend on the current state of the process. From its transition rates and Chapman-Kolmogorov di¤erential equations, we can determine the probability distribution at any …xed time of the process. Conversely, given any probability distribution on the natural numbers, it is possible to determine uniquely a sequence of transition rates of an extended Poisson process such that, for some instant, the unidimensional probability distribution coincides with the provided probability distribution. Therefore, we can conclude that extended Poisson process is as a very ‡exible framework on the analysis of discrete data, since it generalizes all probabilistic discrete models. We will present transition rates of extended Poisson process which generate Poisson, Binomial and Negative Binomial distributions and determine maximum likelihood estima- tors, con…dence intervals, and hypothesis tests for parameters of the proposed models. We will also perform a bayesian analysis of such models with informative and noninformative prioris, presenting posteriori summaries and comparing these results to those obtained by means of classic inference.
Keywords: Processo estocástico; Simulação; Inferência bayesiana; Modelos discretos; Stochastic processes; Extended Poisson process; Chapman-Kolmogorov diferential equations; Discrete models; Poisson, Binomial and Negative Binomial dis-tributions; Maximum likelihood estimators, confidence intervals and hypothesis test of parameters; CIENCIAS EXATAS E DA TERRA::PROBABILIDADE E ESTATISTICA::ESTATISTICA