On weighted Poisson distributions and processes, with associated inference and applications

Abstract

In this thesis, weighted Poisson distributions and processes are investigated, as alternatives to Poisson distributions and processes, for the modelling of discrete data. In order to determine whether the use of a weighted Poisson distribution can be theoretically justified over the Poisson, goodness-of-fit tests for Poissonity are examined. In addition to this research providing an overarching review of the current Poisson goodness-of-fit tests, it is also examined how these tests perform when the alternative distribution is indeed realised from a weighted Poisson distribution. Similarly, a series of tests are discussed which can be used to determine whether a sample path is realised from a homogeneous Poisson process. While weighted Poisson distributions and processes have received some attention in the literature, the list of potential weight functions with which they can be augmented is limited. In this thesis 26 new weight functions are presented and their statistical properties are derived in closed-form, both in terms of distributions and processes. These new weights allow, what were already very flexible models, to be applied to a range of new practical situations. In the application sections of the thesis, the new weighted Poisson models are applied to many different discrete datasets. The datasets originate from a wide range of industries and situations. It is shown that the new weight functions lead to weighted Poisson distributions and processes that perform favourably in comparison to the majority of current modelling methodologies. It is demonstrated that the weighted Poisson distribution can not only model data from Poisson, binomial and negative binomial distributions, but also some more complex distributions like the generalised Poisson and COM-Poisson.