Contributions to count data modelling with applications

Abstract

Count modelling has become increasingly popular over the years. From a distributional point of view, the Poisson and Negative Binomial (NB) distribution are by far the most popular distributions, probably due to their simplicity and historic use. Count data usually exhibits excessive number of zeros, and subsequently zero inflated models have been proposed. However, it is common to have simultaneous excess of counts which do not have to be consecutive integers as most literature supports. In an attempt to overcome this, but also explore other less popular count distributions, we propose three novel Multiple Inflated (MI) distributions. We show with a particular real data how these give superior fit to models already discussed in literature. We further extend this to MI regression models and give a real life example on this too. On another note, there is vast support of univariate count time series models, and multivariate extensions are rather sparse. We use a copula approach to extend some univariate extensions of the popular INGARCH models to bivariate models, and use the Inference Functions Method (IFM) for estimation purposes. We show by means of an example that our construction improves model fit and predictive accuracy as compared to another fit from literature. Furthermore, multi inflation is also present in time series of counts. Again, literature on such models is rather sparse. We propose two novel residuals driven ARMA Zero and One Inflated (ZOI) models, with the Poisson and NB distribution as parent distributions. We derive statistical properties of these and use the partial likelihood function for estimation purposes. Simulation studies confirm that the partial maximum likelihood estimators (PMLEs) behave similarly as regular Maximum likelihood estimator (MLE)s. We also apply our models to a popular data set. To make a step further, we extend these to bivariate models, again via copula approach and use the IFM for estimation purposes. Simulation studies show expected trends and satisfactory results.

Date of Award	15 Apr 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Jingsong Yuan (Co Supervisor) & Georgi Boshnakov (Main Supervisor)